The Prime Minister's comments follow a television interview yesterday with Doreen Lawrence - mother of the murdered black teenager Stephen - in which she said it was unlikely that the family would ever see justice done for their son.
The Prime Minister, who last week praised the courage and tenacity of the Lawrence family, admits Britain lags behind the United States, where Colin Powell was able to rise to the top of the US military, in spite of racial problems there.
"There is no parallel in today's Britain. We still have a mountain to climb before we have a decent, modern, multicultural society we can all be proud of. I want to make Britain a beacon to the world in race equality," he says. "In all honesty, I don't think I can say that enough has been done to improve race relations in the UK."
His remarks in New Nation, a newspaper for the black community, will be reinforced today in a speech by Jack Straw, the Home Secretary, to senior police officers. He will tell delegates at a policing seminar in Gloucestershire that they face a "watershed" after the Lawrence report, and today's publication of a report by the Chief Inspector of Constabulary, David O'Dowd, criticising police forces across the country for failing to improve race relations.
"We are now at a watershed in police and community relations," Mr Straw will say. "This is the time for a permanent and irrevocable change."
It appears increasingly unlikely that the Government will be prepared to implement all 70 of the recommendations in the Lawrence report. There is deep anxiety about the idea of retrying suspects who have been found not guilty, and there is scepticism about the suggestion that privately expressed racist views could be made a criminal offence.