Before Oxford's Ashmolean Museum embarked on its £61 million redevelopment, it was sometimes hard to tell if it was even open, admits its director Dr Christopher Brown. Its grand main entrance doors remained shut - entry was via a smaller side door.

What a contrast with the museum today. Following Rick Mather's splendid six-floor extension and reworking, the Ashmolean presents a far more open, welcoming face. Its grand doors are once more thrown open, the lobby is now step-free and accessible to buggies, wheelchairs and all. The museum stays open till late - 10am on some days. It has a rooftop restaurant to generate revenue to help running costs, plus education and conservation facilities. Temporary exhibition rooms allow it to supplement its main collections with travelling shows. In all this, it's very much the epitome of a 21st century museum. The trick has been to carry out this transformation in sympathetic tandem with the museum's Grade 1 listed 19th century original building, and with its formidable academic pedigree intact - unlike many museums, this is a teaching facility with students taught directly from the objects in the collection.

For the Ashmolean has a truly impressive history. Established in 1693, it can claim to be the world's first ever public museum. It began with the natural history specimens collected by the gardening pioneer John Tradescant and his son, also John. It's come a long way since then. The founding collections were supplemented with artefacts collected by the antiquarian Elias Ashmole and displayed at the Tradescants' home in Lambeth before being donated to the University of Oxford in 1677. In 1845, the collection was installed in new premises built onto the rear of the University Galleries, designed by CR Cockerell to house the university's art collections.

By the early 21st century, the Ashmolean had merged with the art collection and acquired many further artefacts to become an important and wildly eclectic collection of art and archaeology - from drawings by Raphael to Indian, Chinese, Japanese and Islamic art.

Something so disparate needed a strong architectural vision to hold it together, and that's what it got from Rick Mather Architects, which has been working on the project for 10 years. When Mather was brought in, the Ashmolean was desperate for more space and to be able to meet the standard of facilities expected by 21st century museum-goers.

But there was a problem - the presence of lots of inadequate museum buildings at the rear which made expansion difficult. What's more, planners were keen that any new structure had to be invisible from the grand front entrance, a stance that Mather's already held.  After considering and discounting various renovation and excavation options, Mather's answer was to demolish 60% of the museum - everything but the imposing, neoclassical Cockerell building, which was deemed the only structure of architectural distinction, and build out anew at the back.

What it needed, says Dr Brown, was not pastiche but a structure 'that complemented Cockerell's 1845 building using the language of today'.

That's exactly what Mather delivered - a 10,000sq m, concrete-structured extension that seamlessly merges with the existing, deceptively shallow building. As you walk through, you aren't immediately aware of where one ends and one begins. The 39 new galleries double the display space and allow the gallery to exhibit hundreds of objects previously languishing in storage. These include textiles, which due to environmental controls in the new spaces can be shown for the first time.

The outwardly near-invisible nature of the building ensured that the extension avoided the iconic aspirations of many other museum developments.  And perhaps this is for the best all round, allowing more attention to be paid to the sequence of galleries and to the visitor's progression through the space.

Mather's design solution was to create six floors around two, top-lit atria with a rhythm of 3m single and 6m double height spaces that reflect the 6m high Cockerell galleries.  The double height galleries are arranged in a horseshoe shape to meet the rear of the old building, with single height galleries beyond the atria. Mezzanine bridges through the galleries ensure circulation at all levels.

The stunning main atrium space is on an axis with the entrance and is the dramatic visual highlight of the extension. The architect has created an elegant, gently curving staircase that zig-zags from the basement up one side of the space right to the upper galleries and the fourth floor restaurant and roof terrace. The position of the grand staircase helps to emphasise this key main feature, and maximises a feeling of generous space and light.

Views are important in the new building, and were crucial to its design development. Mather planned the gallery spaces to engage with each other through views across the main atrium and, radically, through double-sided display cabinets built into the deep walls, some of around 430 new bespoke cases. The effect is tantalising, giving glimpses that encourage the visitor to carry on through the museum, and ensuring the content shines through the architecture. This legibility was important for another of Mather's design priorities - clear orientation. He was adamant visitors would never encounter dead-ends, as could happen in the museum's previous more labyrinthine state.

'One thing I knew was that I didn't want to get lost in it. I used to come here and not know where I was,' says Mather.

As well as incorporating display cases, these 'fat walls' work hard - they provide all the servicing and ventilation to avoid the need for false ceilings and enable Mather's 6m and 3m gallery arrangement.

The architect also had to solve a thorny problem in the entrance, where the floor first rose three steps into the 'tribune' lobby of fluted columns and then dropped back down again. Such awkward levels changes had to be resolved - today's museum visitor requires full access at the main entrance - but without destroying the attractive setting. Mather's solution works brilliantly - the original floor has been relaid and dropped down to the lower height to give level access. A new plinth below each of the retained columns brings them down to the new floor level, which blends in very successfully.

The palette throughout is simple and classic - oak floors, steel balustrades, glass bridges, polished plaster on the atrium wall, and where old meets new, Portland stone from the same quarry as that used by Cockerell. The effect is crisp, fancy-free and timeless.

The all-important views into the various galleries contribute to another key aspect of the redevelopment - a cross-disciplinary display strategy that creates cross connections across the collections to emphasise cultural closeness rather than cultural differences. These were created by Metaphor, the museum and heritage consultancy led by architect Stephen Greenberg.

The redevelopment was also an opportunity to tackle security - on Millennium Eve, a Cezanne painting was stolen via a skylight. Shutters covering the atrium and steel mesh in outer and inner walls are designed to prevent any further such incidents.

It's been a long process, but the result is one very happy client. At the launch, University of Oxford vice-chancellor Professor Andrew Hamilton eulogised the 'magnificent facility' that allows it to open up its treasures and engage its audience in a 21st century rather than a 19th century style.

"I believe it Rick Mather's finest building to date, and I have no doubt it'll be recognised very soon as one of the outstanding museum buildings of the 21st century ," added museum director Dr Brown, who expects visitor numbers to rise from 400,000 to 500,000 per year. Refurbishment of the Cockerell building galleries will follow in 2011. There is also a further phase to incorporate and re-use a light well to the west of the building.

And neatly, says Dr Brown, the extension brings things full circle. Archives quote early visitors to the original museum as saying they felt like they'd been around the world in a day. Now, with the new arrangement of displays, visitors can say that again.

As well as taking pride in its illustrious past, Rick Mather's impressive new addition to the Ashmolean ensures that it can have a great future as well.