The redevelopment of Oxford's Ashmolean Museum cost £61m. Where has all that money gone? Looking from the street, it's hard to believe a single penny has been spent on this glorious early Victorian building that contrasts so strikingly with the medieval and neo-Gothic buildings it faces. The Ashmolean – an architectural wonder by Charles Robert Cockerell that fuses ancient Roman, Greek and English baroque design – seems all but unchanged since the day it opened in 1845.  This, though, is part of the conjuring trick Christopher Brown, director of the museum since 1998, and his architect Rick Mather have pulled off. While Cockerell's facades have been spruced up a little and the magnificent central door once again forms the museum's principal entrance (a side door was used for many years), those millions have been spent almost entirely out of sight.  It is behind Cockerell's neo-classical facades that a new world opens up, one that radically transforms the scale of this, Britain's oldest public museum. Beyond the Grecian entrance lobby, Mather's unexpectedly large six-storey extension might easily be considered the tail that wags the Ashmolean dog, were it not so restrained and refined. Mather has shoehorned a huge, modern concrete-and-glass box into the courtyard behind Cockerell's creation. From the surrounding streets, this generous extension is invisible; the streetscape, a heady mix of medieval, Georgian, Regency and Victorian Oxford, is untouched.  By any standards, this is a clever sleight of hand, since Mather's extension actually doubles the museum's display space. Its six floors – one underground – boast no fewer than 39 new galleries, including four for temporary exhibitions, together with an education centre, offices and Oxford's first rooftop restaurant. Yet, for all its space, clarity and light, this is a complex design. The galleries are set on two different axes (north-south and east-west) over the six floors. Some are three metres high, others six. They come together like a multidimensional jigsaw puzzle, the pieces connected by enclosed glass bridges passing through and over a great central stairwell, lit from above. This stairwell is itself a quietly spectacular space, one side of which is given over to a gleaming white architectural cliff into which are set flight after flight of white Portland stone stairs, linking the tiers of galleries and the rooftop restaurant. While the geometry is intricate, the effect is relaxed, engaging and generous.  "The idea," says Mather, "is to entice visitors to make a great figure of eight – up, down and through the galleries. The contents of each can be glimpsed from the one before, through openings and windows. So you get pulled along. And, wherever you go, whatever you're looking at, the central stairwell, which holds the design together, is easy to find. We know a lot about 'museum fatigue', so we've made a sequence of spaces that, hopefully, will keep this to a minimum."  Wherever they walk, on whatever crisscrossing floor or bridge, visitors will circle back to that six-storey stairwell, only to be lured into ever more galleries, each presenting more of the Ashmolean's rich collection than has ever been seen before. "Much of what we have has been in store for donkey's years," says Brown, the Ashmolean's 17th director. The first was Robert Plot in 1683, when the museum was housed in what is now the University of Oxford's Museum of History and Science, a building commissioned for the celebrated antiquary Elias Ashmole's collection of curiosities. "Our problem was that the Cockerell building, although very fine, is actually tall and narrow. It's not that roomy.  "In the early 1890s, to get more space, the courtyard was filled with five pitched-roof sheds, rather industrial affairs rushed up by the local architect HW Moore. These were never satisfactory. A part of one shed's roof blew off a few years ago. They were, though, listed Grade I, because they were seen as part of the historically important Cockerell design. So they might have stood in our way. Luckily, the conservationists let us off lightly when we proposed demolition. Once we managed to convince everyone of what we could do with the space, without it being visible from the surrounding streets, we were up and running."  In fact, Brown has been working on the project for 11 years. But then his is a complicated museum, one that houses collections of both art and archaeology. Here you will find Constable cloudscapes and pre-Raphaelites, along with paintings by Dutch masters and Picasso. Balancing these are Greek and Roman sculptures, fragments from Minoan excavations, Middle Eastern robes and even a wooden doorway from Yemen donated by Lawrence of Arabia.  Then there are Anglo-Saxon jewels, prized majolica pottery, and the famous "Messiah" Stradivarius violin no one is allowed to play ("A condition of its being here," says Brown. "Rather sad, don't you think?"). Guy Fawkes's lantern (yes, the very one he carried with him on the gunpowder plot) is there, too, as well as antique bowls from Iraq and Iran, and Chinese political posters from the Cultural Revolution . . . and so the list goes on. The only things missing are a Cézanne, stolen (probably to order) from the Ashmolean in 1999; and what was one of the museum's greatest treasures – the world's only stuffed dodo. Had it not been allowed to rot back in the 18th century, it would have been quite a crowd-puller today.  The genius of 'fat walls'  When it reopens on 7 November, the new Ashmolean should certainly engage a wider public than before, although Brown is cautious when discussing visitor numbers. "Best not to get carried away," he says, suggesting a yearly figure of 500,000, which would be an increase of 100,000. It seems a conservative estimate: this is the first major modern museum in Oxford, a city of wealthy yet small independent colleges that rarely join together to shape a big building for public events.  While Mather's Ashmolean addition is a magical combination of cool stone, oak floors, spruce plywood, polished plaster, steel, glass and zinc, all its elements have been brought together with a lightness of touch. This is a characteristic of Mather buildings, from his Zen restaurants of the 1980s to the new Towner Gallery in Eastbourne, but here it is also down to the way the architects and engineers have tucked the mechanical and electrical services into the walls. So floor-to-ceiling heights are as generous as they can be, while what Mather calls his "fat walls" double as recessed exhibition spaces. The result is a building in which every last inch is hard at work, while giving the opposite impression.  Brown and Mather have not only brought fresh life to the Ashmolean; they have also given Oxford a fine new public place where visitors can meet, eat and while away rainy days rummaging happily through one of the country's great treasure chests. The fact that this enchanting museum is also an active seat of research and scholarship only adds to its lustre, while the reality of seeing so many objects – squirrelled away for too many years – out on display will make the Ashmolean a museum to return to, time and again.
It's Britain's oldest museum – and it's just had a thrilling revamp. Jonathan Glancey gets a sneak preview of the all-new Ashmolean