At first glance, a visitor to Oxford standing in Beaumont Street would struggle to discern the new Ashmolean, a fact that amuses its architect, Rick Mather. Although half the museum is now an entirely new building, and more than £60m has been spent provided largely by the Linbury Trust and the Heritage Lottery Fund, the extension has been ingeniously dropped within a space defined - and in effect concealed - by seven other contiguous structures.

Although the museum has its origins in the collection presented to the university by Elias Ashmole in 1677, in its present form it is largely a late-19th-century creation - indeed. Its very name dates back only 100years. Its use has evolved too; within living memory, drawing classes for students a t the Ruskin School of Art were held in what is now the museum's shop. The main building, by C.R. Cockerell, was opened in 1845 as the home of the university's art collections, including classical sculpture and casts. But archaeology was becoming a serious academic subject and in 1878, 132 senior members of the university signed a proposal for a new museum that would combine the university's art and archaeological collections. In the late 19th century a memorable warren of shed-like galleries was added behind Cockerell's building under the direction of Arthur Evans, the excavator of Knossos, and in 1908 the whole was renamed the Ashmolean Museum of Art & Archaeology.

After being closed for nearly a year, the Ashmolean reopens to the public this month to reveal that those late-19th-century galleries have vanished, to create the site of the new extension. So there is a real surprise - and a lift of the heart - as one enters the Cockerell building to see the transformed interior that opens up beyond. Perhaps the best way to understand Mr Mather's achievement, and to see quite how tight the site is, is to take the lift up to the most unexpected part of the extension, the roof-terrace restaurant, to enjoy a vista of Oxford's golden spires and gables. At night this will be a magical experience, impossible anywhere else in the city - indeed, it is a feature so novel that it triggered some resistance at the early planning stage. An engaging touch, typical of the thought evident throughout, is a neat lawn at roof level.

Although this awkward site offered such a challenge to the architect, engineers and contractors, it has been triumphantly exploited and turned into an asset in the 39 new galleries that the building provides. Small, slit windows allow glimpses out to brick walls, or a back alley, subtly reassuring visitors that there is a daylit world nearby and also helping with orientation. A key element here, as he readily explains, is an experience in Mr Mather's architectural formation half-a-century ago. As a young American in London, he enjoyed the tours of Sir John Soane's Museum in London by its curator, SirJohn Summerson. That is another idiosyncratic building, far larger inside than its exterior would suggest, and a Zoo-year-old exemplar of how to bring light down into a cramped city site. Also, the Soane achieves communication with the minimum of words. Arguing that signage is badly done in most museums, Mr Mather explains that he wants visitors to the Ashmolean to be reassured and guided without intrusive signs. Over the past 15 years he has built an impressive portfolio of museum projects, from the Dulwich Picture Gallery and the Wallace Collection (on another cramped urban site) to the extension of the Virginia Museum of Fine Arts, due to be unveiled next May. He takes pleasure in creating spaces that are calm yet offer a sense of discovery, attributes in which the new Ashmolean is rich - curiosity draws one around corners, or across bridges (although I wonder how easy it will be to pass through the large, solid wood double doors that separate most of the smaller galleries).

Double-height glass doors form the main entrance behind Cockerell's columns and admit a stream of daylight that leads the eye through to the new building. From here, the main staircase, which connects all three Roars of the extension, curves elegantly upwards. Its Portland stone and polished plaster gleam in the light from a glazed roof, a characteristic Mather feature. The main galleries are not the conventional discouragingly large boxes, but are subtly articulated with curving walls, internal windows, and projections for doorways or service areas. Double-height galleries alternate with two-storey spaces that offer smaller spaces for specific themes. Some of these smaller galleries are linked by an internal bridge across the double-height spaces, openings offer glimpses down from one gallery into another.

This interlocking of heights and spaces, which may seem confusing in the plan, opens up vistas or hints at further delights, so that visitors are encouraged to explore. Even within the first-floor galleries, glimpses of the sky are omnipresent, although the design ensures that virtually no daylight falls anywhere neat vulnerable works of art. At the extension's western end, a second staircase, with windows onto St John's Street, sweeps up through three floors, its green-washed wall dressed with plaster busts from the Ashmolean's collection of casts from the studio of Sir Francis Chantrey, for long in storage. When it opened in 1683 the Ashmolean was the first public museum in Britain. From the start a scientific, research and teaching facility for the university, it has changed its focus several times, as collections came in, and either fell from intellectual fashion or were forcibly removed by stronger university institutions - the cabinet of coins and medals, for example, was moved to the Bodleian Library in 1858. This complex story, described by a former senior assistant keeper, Arthur MacGregor, in his history of the museum, is explored in the new extension's engaging gallery UK to Ashmolean'. Much-loved early-Stuart objects from the founding 'collection, including Powhatan's mantle and Guy Fawkes' lantern, are on show again. An engraving captures a moment in the museum's history: it depicts a giraffe in front of a classical door frame over which bangs a painting of Ashmole's stuffed dodo.

A 1711 quotation nearby captures the disgust of a Swiss traveller, horrified by the fact that women were not only allowed in the museum (for 6d) but were also prone to 'fumble' the specimens. The display explains how this cabinet of curiosities became a proto-scientific institution, but suffered intellectual stagnation. In the late 19th century)' those natural history specimens that (unlike the dodo) had survived, and the ethnological collections, were relinquished to the newly founded Pitt-Rivers Museum.

Scholars can be possessive of their specialist areas. Thanks to the tactful and encouraging leadership of its director, Christopher Brown, the Ashmolean's departmental boundaries have been breached and the intellectual shape of the displays enriched through active interdisciplinary collaboration. Despite having a small academic staff, the museum has an outstanding reputation for academic enquiry. 30 years of research into its history have stimulated valuable studies in the history of collections and the history of ideas as well as the history of art, both eastern and western.

Consciousness of historical context is one of the core attributes of the Ashmolean - which is not surprising given the intellectual calibre of its curators, most of whom are teaching members of the university. A sense of local associations, linked to an international context, is a theme that runs throughout the museum. Geographical settings are constantly evoked. In the Mediterranean World Gallery, a huge map, some 18 feet long and lit from below, defines the Mediterranean world as stretching from the Himalayas to Oxford Elsewhere, a beautiful drawing of mountain peaks in Asia is reproduced as a backdrop to ceramic figures of Bactrian camels laden with packs of luxuries from China with fragments of coloured textiles and shards of porcelain helmv. Even without text, the message of long-distance trade along the Silk Road, and its geographical difficulty, is explicit. Some of the exhibits are both large and surprising, such as the 300-year-old 10-feet-tali carved doors from an Arabian merchant's house, probably carved by Indian craftsmen, that were brought back from Jeddah by T.E. Lawrence in 1921, a highlight in the Asian Crossroads Gallery. Knowledge lightly borne and expressed in simple, telling language is always a pleasure to encounter. In the Money Gallery, for example, direct comparisons of buying power illuminate a 4,000-year timeline, so that we learn how much silk cost in 10th-century Alexandria, or what Marco Polo paid for salt in Chinese paper notes in 1277. Overall, there is plenty of variety in both colours and graphics, within the template supplied by the design consultancy Metaphor. In each display, striking objects are sited so as best to catch the eye, with few uniform ranks of objects. Throughout, supporting graphics, colours and quotations allow the past, to some extent, to speak for itself reinforcing messages and insights through all the senses. Unlike the repetitious displays and stark presentation considered orthodox today in many art museums, which rely too heavily on graphic panels and a standard authorial voice, here there are many voices, tones and nuances, showing how well the design team has drawn on the wide academic and educational expertise of the museum's staff.  For architectural students, museologists and others intrigued by the thinking behind the new Ashmolean, the process of its creation is explored in a temporary gallery that looks at every aspect of the project from the choice of stone to the design of graphic panels, a refreshing generosity of approach.

The original galleries for western art in Cockerell's building have been refurbished, relit and rehung, so that they too can be enjoyed afresh. In the Dutch Golden Age gallery, for example, a central case of colourful and massive tin-glazed vessels draws the viewer, who then swings around to take in the Old Master paintings that line the walls. Hundreds of artefacts have been put on show for the first rime, drawn from the half million or so owned by the museum. A third more material from the eastern art collections is on show, including a gallery and study room for vivid and fragile Islamic textiles, drawn from what is reputedly the best collection in the world.

Philippa Glanville

On the extension's lower level, the conservation department has devised three new galleries. 'Restoring the Past' looks at the history of conservation from the ancient world until the 19th century (Fig. 11). In 1895 Richard Young joined the museum as one of its first restorers. When mending Greek pots, he signed his surname in Greek, 'Neos' , as the ancient potters did, followed by the date, turning the good museum practice of identifying restoration into something witty and characteristic of this idiosyncratic institution. In the second gallery, 'Conserving the Past', uncertainty about dating is exploited to good effect: a bronze arm is described as either Greek or Roman, or 'even of the 1500s', setting up a question in order to show how scientific research techniques might offer an answer.

'Exploring our Past' includes some more formal science, looking at materials and how to analyse them. Striking objects and graphics catch the eye, and the texts are large and clear, so that insensibly one absorbs detailed explanations of research methods. Behind the scenes, a large new loading bay and service area will be essential for bringing in art for the ambitious series of exhibitions planned over the next five years. New studios mean that the conservation department can now treat on site almost all types of object, including sculptures, paintings, works on paper and textiles. Thanks to these new resources and a suite of four galleries designed for temporary exhibitions, international partnerships are feasible. In September 2010 the Ashmolean's first major show since its reopening will be 'The Pre-Raphaelites & Italy', a collaboration with the Museo d'arte della citta di Ravenna. This will be followed in 2011 by a spectacular archaeological show; on the royal tombs at Vergina in Greece. Smaller exhibitions, collaborating with Oxford institutions and regional museums, are also planned.

Intelligence and wit run throughout the displays. It is clear that the specialist curators have been encouraged to give expression to their particular enthusiasms in the design - for example, the deep red and ochre walls and case linings for the Indian Gallery, chosen by Andrew Topsfield, the Keeper of Eastern Art. This is in line with the architect's simple aim, 'that dle objects should look better than ever before', an ambition that has been brilliantly achieved.

Philippa Glanville is a former keeper of metalwork at the Victoria & Albert Museum, London.