Future classic: The world's first public museum gets a £61m makeover

Oxford's Ashmolean Museum of Art and Archaeology re-opens on 7 November, after a £61m transformation, Britain's biggest cultural project since the creation of the £100m Great Court at the British Museum. The Ashmolean's extension doubles its display space with a pale, luminous architectural clarity that puts the flaccid classical pastiche of the adjoining Sackler Library very firmly in its place.

The new six-level building is an object lesson in how intelligent architectural contrast can refresh the sacred cows of our cultural landscape. Indeed, the design of the Ashmolean's extension recalls the recent remarks of Andrew Hamilton, the Oxford University's new vice-chancellor, who was previously provost of Yale. "I was expecting to find a place somewhat bound by the past," he said. "Actually, what I've discovered is a place hurtling towards the future, and I would give as examples not just the commitment to the education of our future citizens but also this emphasis on tackling problems of the modern world... I do not see Sebastian Flyte and his teddy bear as I walk around Oxford."

The Ashmolean may not be exactly hurtling towards the future, but the architecture of its extension celebrates the relationship between new and old architecture in a way that is dynamic, rather than cowed by historic precedent. The Sackler Library looks eerily like a demountable backdrop from Mel Brooks's satirical film, History of the World: Part I; the Ashmolean's extension strikes a finely wrought contemporary relationship with the museum's classically inspired 19th-century architecture, by Charles Cockerell.

"Oxford has a passion for pastiche," said the Ashmolean's director, Christopher Brown. "I wanted a building for today." Behind him, a window cleaner, high up above a case containing 16th-century cithern stringed instruments, squeegeed glazing on the gallery wall in the race to get the new building ready for its opening. The bustle included Ashmolean curators, carefully unwrapping a blanketed catafalque containing a canoe fashioned by Barbara Hepworth, brought in through the museum's first proper loading bay, and into internal conditions which – also for the first time – are environmentally controlled.

And so, more than three centuries after its foundation in 1683, the Ashmolean casts off its legendary physical crampyness with an almost seamless architectural flourish that has created 39 new galleries, with 420 Belgian-made display vitrines costing £7m, in a light-filled architectural cabinet of curiosities designed by Rick Mather Architects.

At its heart is a wonderfully elegant central atrium glowing with polished plaster and a beautifully asymmetric staircase. The ghost of Elias Ashmole, whose moth-eaten dodo was a key exhibit until only its tragic head and one claw remained in 1755, may soon be seen floating through the museum's new spaces trailing a vaporous glimmer of St Elmo's fire.

The Ashmolean's original 17th century building – possibly the work of a young Christopher Wren and Dr R T Gunther – was described in the 1706 publication, The New World of Words, as "a neat Building in the City of Oxford". Mather, a fastidiously neat modernist, has simply upped that particular ante with a virtuoso design that is the architectural equivalent of jJenga, the game in which players remove interlocking blocks from a tower, one by one, creating a delicately poised matrix of pieces.

In Jenga, the tower eventually collapses when one block too many is removed. At the Ashmolean, Mather and his project architect Stuart Cade have created a matrix of solids, voids, bridges, crisply detailed linking galleries, and subtle shifts of lighting in a realm whose vistas penetrate one another in enfilades and spatial overlays.

Standing with the Ashmolean's director on the second-floor landing, the effect is almost dizzyingly panoptic. From here, one can see Islamic and European ceramics, traded objects from the Far East, a Samurai suit of armour, 19th-century Meji pottery, and part of an ornately carved door snaffled in India by T E Lawrence. This collage effect is rampant almost everywhere, and so is the sense of connection. In a single glance, one typically glimpses at least four exhibition spaces and two bridges crossing at various levels. The architecture, switching from galleries either three or six metres high, blurs scholarly territories, supporting the museum's new display strategy: Crossing Cultures, Crossing Time.

It's tempting to see all this as a perfect stage set for Jacques Tati's bumbling Monsieur Hulot, in a film which he might have titled "Jour d'art". But it is also going to be a wonderland for visitors to the Ashmolean, which was the world's first public museum. The numbers are expected to increase from 400,000 to 500,000 a year, but this is surely an underestimate: the extension's going to be a mecca for school parties in particular. And its pulling-power to scholars will also increase, now that key collections and individual items can at last be properly shown: a cache of Byzantine icons, for example, Henry VII's cloth-of-gold funeral pall, Gandhara Buddhist relics, vastly more archaeological material, and modern art in ideal viewing conditions.

It is perhaps telling that it has taken a thoroughly modernist design solution to achieve this. And it is a testament of Christopher Brown's and Rick Mather's shared vision of bravely new old worlds that the Ashmolean experience would have re-infuriated the patrician German visitor who, in 1710 deplored the presence of "ordinary folk" in the museum– people who "impetuously handle everything in the usual English manner, even the women are allowed up here; they run here and there, grabbing at anything." Or lunching in the Ashmolean's chic new rooftop restaurant.

Oxford's Ashmolean Museum of Art and Archaeology opens 7 November (01865 278000)