The Cabinet decision, which will be announced in a Government White Paper next month, ends a battle between the Foreign Office and the Home Office over extending rights of citizenship status to 13 British Dependent Territories.
The Independent has learned that the Foreign Secretary also overcame stiff resistance from Alistair Darling, the Social Security Secretary, over the possible increased cost in the welfare budget if an influx of fresh immigration is allowed. Mr Cook got the backing of the Cabinet after arguing that few would take up the right to enter Britain.
Those who will be granted rights of access occupy the "last pink bits on the map", including around 60,000 inhabitants of Bermuda, the Caymans and the British Virgin Islands, who are thought least likely to see a future in Britain. Around 10,000 islanders from Montserrat made homeless by the island's volcano have already been given the right of abode to escape the catastrophe.
The White Paper will restore automatic rights to settle in Britain, which were withdrawn in 1962. When the British Nationality Act came into force in 1983, 11 of the colonies were left without full citizenship. The occupants of two disputed territories - the Falkland Islands and Gibraltar - retained their status. Pressure for full citizenship has been strongest in St Helena, the South Atlantic island which is Britain's second oldest colony after Bermuda, with about 5,800 inhabitants of mixed origin.
The relatively peaceful handover of Hong Kong to the Chinese convinced the Home Office that it was unlikely to rekindle the demands for full British citizenship for up to three million people who were given reduced rights as British Overseas Citizens (BOCs) under the last Conservative Government.
A spokesman for the Joint Council for the Welfare of Immigrants said: "Jack Straw has been very sensitive to the fact that the Home Office has opposed the principle because of its concern about granting British citizenship rights to Hong Kong people. It was only prepared to go down that route if the Foreign Office agreed to take the lead, and take the flak. They are prepared to do that.
"The [Foreign Office] has been getting a lot of its time taken up by receiving lobbies from microscopic islands and from big-wig supporters. They are quite successful at getting MPs and peers who are - quite properly - concerned about their rights. There is no prospect of any influx of these people coming into this country."
The Government is anxiously watching the outcome of a case which was referred recently by the High Court to the European Court of Justice for BOCs to have rights of movement across the European Union. If their appeal is upheld, it could reopen the whole question of the citizenship claims by the former Hong Kong nationals.