Oxford familiars over the years have got to know the Ashmolean as a forbidding place. In muscular early Victorian classical style, it was a monument to severe learning and a demonstration of the university's vast, unsmiling pride. Designed by CR Cockerell, who spread the high seriousness of Sterling throughout the country with his provincial offices of the Bank of England, the Ashmolean presented a windowless Greek wall to Beaumont Street. Relief came only from pilasters.

For many years the symbolic entrance under the massive portico (with its capitals fastidiously copied from the Temple of Apollo at Bassae) was shut. One had to use a side entrance, rather as furtive tradesmen, to access the Stygian and cluttered collections. And while it was not official policy to discourage visitors, the grim Ashmolean certainly intimidated them. To enter was to breach the protocols of a club privée. A visit for pleasure was as gross an intrusion as taking a whoopee cushion to high table in Magdalen.

That has now changed, and rather radically so. The magnificent facade and portico remain intact, but an entire new museum of 39 galleries and about 100,000 sq ft has been built, almost surreptitiously, behind. The architect is Rick Mather, an experienced American Londoner for whom the term "rangy" might have been coined. His other recent museum work includes the extension to Dulwich Picture Gallery and the Towner in Eastbourne, each rightly admired. The first thing he did was create a proper entrance where a proper entrance should always have been: in the portico. An apologetic door has become a huge double-height glass feature: a tribute to the semantics of welcome and the design of change.

The Cockerell building of 1841-5 was a conceptual oddity: its blustering facade hid only a fillet of space one room deep. It was a Potemkin village of a museum: an authentic, long "gallery". The only exception was an apse on the axis of the portico, demolished in the 1890s to be replaced by industrial sheds undistinguished in function and form. The removal of these sad sheds created the opportunity for the cheerful new building.

Mather wanted to make sense of Cockerell's original design. The old apse gave an important illusion of depth which has now been restored. Visitors get a splendid, inviting vista into the new building : dead-ahead on line-of-sight is a distant and vast plaster cast of the Apollo from Olympia. It stands, rising through nearly two storeys, in an atrium which is the central dramatic space and the defining feature of Mather's polite and well-considered design. It's a real tutelary deity in what is still an academic department of Oxford University.

This is not merely a big extension: the new building allows an entirely new reading of its superlative collections. Hitherto hugger-mugger in costive curatorial zones, with little attention paid to matters of interpretation or delight, the museum's five departments are now spread over six floors (one for each plus space for temporary exhibitions). About a third more has come out of store and gone on display, but the important innovation is how the architecture stimulates access : the atrium and glass balustraded walkways "help reveal the whole museum", Mather says. There are always internal views which invite curious exploration but there are external vistas too: the sky and the city help turn the dark and claustrophobic old Ashmolean inside-out.

The original museum of 1683 was based on the collections of Elias Ashmole, alchemist and antiquarian, a leading figure of "The New Philosophy". It was literally a "cabinet of curios", including a Dodo, artefacts acquired from credulous Native Americans and hand-me-downs from the Tradescants. Ashmole explained that his purpose was to encourage "the inspection of particulars... extraordinary in their fabrick". By the early 18th century it was already a busy popular museum, although one sniffy German visitor of 1710 said "Herr Burgermeister Reimer in Luneberg has twice as many specimens".

Since Ashmole's exotica were merged with Oxford's own collections in 1908, that shortcoming has been addressed. Now the Ashmolean is pretty much uncontested as the greatest university museum in the world. The task of reinterpreting the unique mixture of archaeology and art, precarious in so competitive an academic environment, was given to Metaphor, an experienced firm of museum installation specialists. The editorial direction is "Crossing cultures, crossing time", intended to disrupt conventional style labels and sclerotic views of culture. It is an intellectual narrative that complements the architectural circulation: thus, the Ashmolean's director, Christopher Brown, draws attention to Gandhara culture which produced Buddhas dressed in Roman togas. I suppose you could call it multiculturalism. Under the old methodology, such a figure might have perplexed both classicists and orientalists doomed by their conceptual provincialism. Such an incongruity now makes sense, although if the scholarly Ashmole's ghost is doing the rounds he may be perturbed to find a reference to My Big Fat Greek Wedding in the text panel that's headlined "What Have the Greeks Done for Us ?" The gods of "access" do, under lottery funding rules, have to be appeased.

Nice touches? The mechanical and electrical services are hidden in wall cavities so as to maximise ceiling heights. The lovely top floor restaurant space is reached, Mather says, as a "reward" after climbing a fine zig-zag of setback stairs. On the ascent you pass a wonderful wall, the full height of the atrium, with Sir Francis Chantrey's busts of worthies hanging strangely in space. Criticisms? None really; the privileged fug of the old has been airily democratised. Cockerell's Etruscan red paint is complemented by Mather's brilliant white. I stood in the impressive atrium and found myself thinking of the Hyatt Regency hotel in Atlanta. But that's just to say that successful modern museums are in the entertainment business. So, a different, and rather wonderful, sort of alchemy has taken place in Oxford.

The Ashmolean reopens on 7 November

Stephen Bayley
The Observer