As the US watched the millennium approach, time zone by time zone, the anchorman for the ABC network, Peter Jennings, watched the fireworks cascading over the Thames and offered his appraisal of recent British history. "This country has been through so much," he said. "In 1900, when Queen Victoria was on the throne, Britain ruled over one fifth of the world's population. But for all this fantastic show, Britain's possessions have dwindled to... Well, Hong Kong has gone now and, well... The Falklands are still British."

There is only one thing more staggering than the pace and scale of the demise of the British Empire, and that is the pace and scale with which the British have sought either to elevate it to the level of myth or denigrate it with historical amnesia. For most of the past 30 years, since the bulk of former colonial territories gained independence, that has been the framework in which most discussion about this vital period in British history has struggled to take part. Caught between those who wished to romanticise the past and those who sought to forget it, there has been limited space to chart a path from our imperial past to our multicultural present.

It is through that narrow space that the trustees of the British Empire and Commonwealth Museum, recently opened in Bristol, have had to duck and dive as they have sought financial backing and political support for the enterprise. Ten years on they have provided a valuable and accessible narrative, imaginatively and carefully told, that helps those who are interested to reconnect with a vital part of their history. "It has been a very fine line we have had to tread," says John Letts, the museum's founder. "Everywhere we have tried to tell both sides of the story."

The difficulties they encountered getting the project off the ground, not to mention completing it, tell us much about why we need such a museum and why we needed it done well. It set out aiming for £20m, with a sizable portion from public funds, such as the lottery; it ended up with £8m, almost all of it coming from private donors.

The plans went through many incarnations. An early proposal to build the museum on the site of the former Crystal Palace, used for the Great Exhibition of 1851, was abandoned. Instead its home is in Isambard Kingdom Brunel's Great Western Railway Station at Temple Meads in Bristol: a monument to the industrial age in a city linked to the Empire through the slave trade and migration.

"Part of the problem was simply reticence about tackling such a controversial topic," says Katherine Prior, the museum's historical adviser. "There is a squeamishness in Britain about Empire and race."

That squeamishness illustrates an amnesiac quality to the way Britain engages with the murkier parts of its own history. The southern states of the US are peppered with museums to the civil rights era and plaques to civil rights battles. They did not come about without a fight, but they arrived far faster and with far more resources than any commemoration of the Empire, whose demise broadly coincided with the end of segregation.

Americans have sufficient confidence in their history to award the Vietnam veterans who stopped the My Lai massacre, when the men of Charlie Company murdered 300 innocent civilians. Admittedly, it was 30 years after the event. But it is difficult to imagine that in nine years we will even remember that the sinking of the Belgrano happened, let alone honour anyone who tried to stop it. None of that makes American history more morally acceptable. But it does make official versions of British history less believable.

The result is a justifiable concern about the motivation of those who wish to evoke difficult parts of our past. Over the past decade, the museum has tried out several names. A reference to the Commonwealth was added and the word "British" disappeared temporarily, only to be replaced.

The author Jan Morris, a patron of the museum, complained: "The word 'empire' itself, we are assured, puts potential benefactors off. Theme parks about dinosaurs or King Arthur, certainly; reconstructions of Viking York or the Age of the Celts, yes; but a great museum about the British Empire - the most thundering contribution these islands ever made to history - well, on the whole, perhaps not."

Well, that depends what you are going to do with that history. Suspicions could hardly have been tempered with the news that the chief backer of the project was Jack Hayward, the industrialist and chairman of Wolverhampton Wanderers football club, nicknamed "Union Jack" for his patriotism. Other donors include the Prince of Wales Foundation, the Rhodes Trust and the Beaverbrook Foundation. One of the trustees was Kenneth Baker, the former Tory education secretary.

"The naive and rather offensive assumption appears to have been that such men would only support a jingoistic and nostalgic survey of empire," says Prior.

On the contrary, it seems naive not to have reservations about such a sensitive project being bankrolled by those who have no record of showing any sensitivity for the subject. But this isn't a reason to reject it summarily, as many government-funded bodies did. The trustees, says Prior, could not win. "They were being judged on a museum that existed only in their critics' imaginations, and that judgment was blocking the support that would enable them to prove otherwise."

Now that it does exist, it is clear that just because suspicions are well-founded it does not mean they will necessarily be fulfilled. Through 24 themed rooms the museum tells the story of the Empire with caution but not embarrassment. What the British have called the Indian Mutiny, and Indians have referred to as the first war of independence, the museum calls the "great rebellion of 1857". It tells us how words such as "bungalow" and "shampoo" came to these shores while some 300,000 white servants and prisoners left them to work on tobacco plantations before sugar and slavery took over. Pictures of childhood friendships and tales of how hypocrisy and racism frustrated interracial relationships combine personal drama with political and economic upheaval.

Leaving the best until last, a series of personal testimonies from a variety of British voices from different races tell their stories. There are tales of economic migration, joy, discrimination, disappointment and integration. The result is a lasting imprint of the economic motor that drove Empire and colonisation and the human impact such a mammoth journey has had on all of us.

· The British Empire and Commonwealth Museum, Bristol, is open daily, 10am-5pm. Details: 0117-925 4980