When I lived in Cambridge during the early 1980s, I can only recall venturing into the Fitzwilliam once. And once was enough: its grand façade on Trumpington Street never tempted me back in. By contrast, my first visit to Kettle's Yard (the other major gallery in town) was an illuminating experience, which was followed by almost weekly returns. Everyone working at the Fitzwilliam to whom I relate this unflattering comparison is eager to reassure me: they have heard its like before.

The museum clearly had a problem with its image and the accessibility of its collections and something needed to be done. The answer for Duncan Robinson (the Fitzwilliam's director) and his team lay in a building project that would reclaim and exploit a space all but lost within the museum's structure: a hidden courtyard. The parallels with other recent capital projects are unmistakable, and metropolitan snobs might be tempted to think that the Fitzwilliam has attempted a provincial imitation of the British Museum's Great Court. This would be unfair.

What has emerged after some three years and £12m of building work, masterminded by John Miller and Partners, are almost 3,000 sq meters of new space, spread over four floors. The project has rather subtly integrated the new elements into the old, enabling what was already there to be thoroughly reinvigorated.

At ground level, visitors who might be daunted by the classical grandeur of the Founder's Building can now slip in through an altogether more accessible entrance and find their bearings in a new orientation space. Below in the basement is a new education suite, which will come into full use in the next academic year. On the first floor, in a mix of new and redesigned spaces, are galleries that will be used to present both temporary exhibitions as well as the museum's eclectic range of 20th century works.

Though never quite magnificent, the layout and presentation of new spaces is neatly configured and elegantly finished. If forced to confess a general reservation about the architecture, I would question whether too much has been squeezed into too small a space.

But this is a minor quibble, when set against a series of more significant successes. For the mix of limestone, glass, pale wood, brickwork and plaster walls really does capture and then enliven an important series of ideas. Three of the biggest that face any museum these days are particularly well resolved in the new Fitzwilliam, one for each of the main floors in the new building.

Starting at the top, the first bold gesture taken by the new building relates to the importance of temporary exhibitions as engines of change and drivers of visitor loyalty. A lively programme of temporary exhibitions can do so much to rejuvenate a museum.

According to Robinson, the programme of planned exhibitions will draw on the collections, the range of curatorial expertise at hand, as well as local research and teaching energy. What the Fitzwilliam now has is an adequate space in which to channel all this, so that the museum can advance its own thinking, but also, crucially, do so within a public arena.

Downstairs in its new ground-floor scheme the museum quite rightly acknowledges another key concept of contemporary museum visitation, namely the mentality with which people today expect to enter a gallery. The somewhat debunked 19th-century idea that it has replaced is, in fact, still on offer 50 metres further down the Fitzwilliam's façade. Here a humbling trudge up a dozen or so stone steps is rewarded by the magnificent, if somewhat bombastic, architectural flourishes of its grand front hall.

Even if it were not for a greater awareness of just how many people found such threshold experiences daunting, if not physically impossible, economics and public accountability have forced museums to make getting into them easier and more inviting. The importance of this shift in attitude about this simple idea of how people get into museums and galleries is reflected in the most over-worked term in the modern museological lexicon: access.

Today the assumption that people would arrive in a mood of awe and gratitude has been almost entirely replaced by the far humbler notion that they might not pitch up at all unless enticed by a mixture of marketing, immediately visible opportunities for retail experiences, and the promise of refreshments.

The new Fitzwilliam seems to provide a text-book solution here, with its open-ended mix of shop and cafe almost visible from the threshold. What adds grace to these entirely sensible and pragmatic, but not quite spiritually uplifting, amenities is the open courtyard into which they spill. It is here, looking up through the glass roof maybe, that visitors briefly, but crucially, have a chance to pause for reflection - to slow down a notch - before setting about absorbing the visual feasts on offer just beyond the hissing cappuccino machines.

The third bold idea grasped by the refurbishment lies in the basement, where a reinvigorated commitment to education is embodied in a substantial subterranean suite: a seminar room for 50 participants, a schools' room and a studio for practical activities.

More than just bricks and mortar, the reinvigorated pedagogical imperative has also resulted in new education staff whose efforts will focus not only on collections-based teaching and lifelong learning, but also on facilitating public use of ambitious IT resources being simultaneously unveiled by the museum. At the other end of the educational spectrum, there are also connections being fostered with the university's teaching and research into the history of art - learning resources for both town and gown.

A test of any building project lies in what it enables people to think and do within the spaces it creates. At its best, the new Fitzwilliam looks set to embrace contemporary visitor needs, while still remaining true to its founder's vision for his museum: to promote 'the increase of learning'.

Ken Arnold is the head of public programmes at the Wellcome Trust

Project data

Cost: £12m
Funders: HLF, Trinity College, Cambridge, the estate of the late Paul Mellon, an anonymous donor, gifts and donations from charitable trusts, foundations and individuals
Design team: John Miller & Partners, Davis Landgon & Everest, Campbell, Reith, Hill, SVM, Gardiner Theobald, Amec, Light Matters
Cases I&Co/Armour Systems Ltd