The Ashmolean in Oxford is one of the great museum buildings of the world and arguably the most sophisticated classical building ever erected in this country. But as a modern museum it fails lamentably.

This is not the fault of the architect, CR Cockerell, whose galleries make a wonderful museum space. But venture behind them and you find a rabbit warren of later additions with none of Cockerell's distinction and every problem known to modern curators.

It is, quite simply, an unworthy setting for a museum whose collections are finer than some national collections - including the largest and most important group of Raphael drawings in the world, the greatest collection of Egyptian pre-dynastic material outside Cairo, the finest Anglo-Saxon treasures beyond the British Museum, and the only great Minoan collection outside Heraklion.

The museum's problems stretch back to the beginning of the last century when a flood of archaeological acquisitions poured in. Large, cheap, essentially light industrial buildings were added to house them behind the Cockerell building.

Hot in summer, cold in winter, they lack environmental controls and are bizarrely built on a slightly different level to the rest of the museum, making easy access impossible.

To make things worse, a mezzanine installed in the principal gallery in the 1970s blocks the obvious vista from the museum's entrance, presenting such an unenticing view that most visitors ignore it and instead climb the stairs to the Ashmolean's outstanding picture collection, missing many of the finest works in the museum.

The problems pile up. At the moment only about five per cent of the Oriental ceramics are on display, and there is nowhere to show the superb textiles collection.

The museum has no proper conservation space, no dedicated education area and no secure loading area, so priceless objects have to be wheeled out on to the street when anything goes out or comes in on loan.

All this is about to change. By late 2008 the Ashmolean will have been transformed, thanks to an ambitious £49 million rebuild designed by Rick Mather. So far, £36 million has been raised, £15 million of it from the Heritage Lottery Fund.

All the later galleries behind the Cockerell range are to come down. In their place will be five floors of environmentally controlled galleries that will double the display space, topped by a rooftop restaurant with views over Oxford's dreaming spires.

A new vista will entice visitors through from the entrance into a double-height gallery and then round a U-shaped loop back to the Cockerell building. There will be a dedicated education area with its own entrance on St Giles's, conservation workshops, offices that will bring the curators together for the first time, temporary exhibition galleries larger than those in the Sainsbury Wing of the National Gallery, and space for the textiles.

A new shop will release a fine gallery back to museum use, while moving the restaurant means that the vaulted basement can also become a gallery - it would be perfect for mummies.

Spatially, Mather's scheme looks intriguing, with a carefully interlocking sequence of galleries, some double-height, some single, all inter-penetrating, with a dramatic staircase to draw light right the way down into the lower-ground floor.

The intention is to make the museum and its collections much more logical and comprehensible to the visitor, but it has the potential to be a fine space in its own right.

"It was Rick's sensitivity in rethinking and extending the Dulwich Picture Gallery that persuaded us he was right," says Christopher Brown, the Ashmolean's director. (Brown was a DPG trustee at the time, and Lord Sainsbury, a key supporter of the project, chairman.)

"This is not about making the Ashmolean like every museum in the world. Dulwich is the model because retaining the historical integrity and character of the museum is so important."

But there will be more to the new Ashmolean than just a new building. The whole way the collections are displayed is being rethought, with exciting implications. As Susan Walker, the head of antiquities, explains, the catchphrase that sums up the new approach is "crossing cultures, crossing time".

The new Ashmolean will be about making connections. "We mustn't see these cultures in isolation but see them interacting. The new displays will allow the visitor to walk from Rome, through Anatolia, the Levant and India to China all on the same level.

One of the most important things is that the Asian collections will be on the ground floor along with the classical collections. At the British Museum the Asian collections are tucked away on the first floor." As Brown points out,

"It's much easier to make cross-cultural connections here than at the British Museum or the Louvre because we have a wide diversity of collections but are not too large.

"The whole point of this is raising the game," he continues. "The collections here are of greater importance than a number of national collections and we could do so much more with them if we had the money and the right building. After all, what is a university museum for? We should be at the cutting edge of thinking about how we interpret our collections."

It is an interesting question because in recent decades university museums have sat rather uncomfortably with the academic institutions they were built to serve. Brown believes this is changing at Oxford, a university that is currently rethinking the whole way it operates. "I think the university has an understanding of the importance of its museum, in the first place for teaching."

Here he cites the new undergraduate art history degree, which not only uses the collections but relies heavily on the curatorial staff of the Ashmolean for teaching. And, as a university building within a city dominated by the colleges, the Ashmolean has a symbolic significance for the university that has yet to be exploited.

A new building and a fresh approach to the way collections such as those in the Ashmolean should be displayed and explained: in 2008, the Ashmolean will at last be worthy of Cockerell's façade.