They believe that the embattled talks - which made little progress last week - can yet result in a treaty to cut emissions of carbon dioxide, the main cause of global warming. And they are already making plans to lead an international drive over the next six months to ensure that it is properly implemented.
On his arrival yesterday in Kyoto, Mr Prescott was immediately asked by the Japanese chairman of the conference, Hiroshi Okhi, to help him broker a deal using contacts made on two world tours that he took in preparation for the talks. He went straight into a series of meetings with representatives of the United States, Australia and New Zealand, the industrialised countries most reluctant to make cuts - and has a special meeting with US Vice- President Al Gore tomorrow.
Back in London, the Prime Minster is in regular contact with President Clinton, several European leaders and the prime minister of India, a country central to the position of developing countries at the talks.
The two men, with Foreign Secretary Robin Cook, are showing an extraordinary degree ofcommitment to the issue and plan to keep the battle against climate change at the top of the international agenda at a series of summits which Britain, by a remarkable coincidence, will host in the first half of next year. These include a meeting of Asian and European heads of government in Cardiff, an EU summit in London and the G8 meeting ofleaders of the main industrialised nations in Birmingham.
During the first six months of the year Britain will also hold the presidency of the EU, and is determined to use its position to push for major cuts in emissions. This amounts to a remarkable change in Britain's image from the "dirty man of Europe" to the industrialised country that is most determined to reduce the pollution that is changing the climate.
Yesterday Mr Prescott said there had been "a lot of shelling" at the talks over the past week and that he would "go down to the line". But he believed that a treaty would be agreed.
He said: "Nobody wants to be blamed for failure. They don't want to be seen as the dog in the manger."
The likeliest conclusion appears to be that different industrialised countries will pledge themselves to their own, differing, legally-binding targets for cutting emissions. That is less than the European Union wants - it has been pressing for a general cut of 15 per cent by the year 2010. But, as it has already agreed to let its own members achieve different targets within an overall EU reduction, it has effectively conceded the point.
Some Third World countries are already suggesting they will voluntarily commit themselves to limiting their own emissions which may go some way to meeting US demands that they must be brought into the treaty. And delegates are working on schemes whereby dirty countries could buy the right to pollute from the cleaner ones or finance clean-up measures abroad.
Mr Prescott said that Britain was determined "to keep up the momentum" at the summits next year and added: "Tony Blair's effort has been very considerable, and he is at the very heart of this agreement.
"He is playing an important part in bringing about this most important convention, and in seeing it implemented."
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