Don't hold your breath, but the embattled Tony Blair may yet leave office with the world's plaudits ringing in his ears. And it is looking increasingly as if he is defying the pundits and staying until June - as he announced last week - precisely because by then there may have been a breakthrough on one of his chosen "legacy" causes.

Not Iraq, of course. Nor his increasingly frustrated attempts to revitalise public services. Nor even, sad to say, debt relief and African poverty. But the issue he has described as "long term, the most important we face" - global warming.

The Prime Minister now believes that the first six months of this year could be the "tipping point" in finally getting international agreement to take serious steps to combat the climate change - including, unlikely as it might seem, from President Bush. And, though I have been critical enough of his failure to match his words with deeds in the past - especially in failing to take on his White House war-mate on the issue - I have to admit that, for once, he does seem to be bringing effective pressure to bear.

The first fruits, as The Independent on Sunday predicted four months ago, will start to show in the Toxic Texan's State of the Union address on Tuesday. But Mr Blair - and a growing band of world leaders - are playing a longer game, to climax at the G8 summit in Heiligendamm, Germany's oldest Baltic Sea resort, in early June.

Angela Merkel - as Chancellor, the summit's host - is the Prime Minister's chief ally. Ministers feared, when she came to power 14 months ago, that she would lower Germany's environmental profile. But her government has been stronger on global warming even than the Red-Green coalition that preceded it. It was predictable enough: as a former environment minister she was one of the architects of the Kyoto Protocol. And, following Mr Blair's example at Gleneagles two years ago, she has put climate change at the top of the G8 agenda.

The Prime Minister pressed the President on global warming when he visited him in December, and they have had telephone conversations on it since. Mrs Merkel followed up when she visited Washington at the beginning of this month. Both will be rallying support this week at the annual meeting of top politicians and business leaders in Davos.

They have already converted Jose Manuel Barroso, who set out, when he became president of the European Union in 2004, to play down the environment. Tony Blair helped to convince him that global warming could regain the EU the popular support it seemed to lose in its constitutional referenda. He pursued the point with George Bush when he went to see him a few days after the German leader, and has since proposed tough measures that would put Europe in the lead in tackling climate change.

Shinzo Abe, the new Japanese Prime Minister, has also joined up and has promised to keep up the pressure when he takes over the G8 presidency next year. The re-elected President Lula of Brazil - a key figure in mobilising developing countries - has signalled his readiness to join the campaign. And Mr Blair is keeping close touch with the leadership of China and India, the other two countries who hold the key to success.

It is all a far cry from only two years ago when the Prime Minister would complain that he was the sole leader who wanted to talk about climate change. To add to the pressure, Mr Bush is losing the support of his two staunchest allies - John Howard, the Australian Prime Minister who was the only other developed country leader to reject Kyoto, and the Canadian premier, Stephen Harper, who has been sceptical about the treaty. Both are now modifying their domestically unpopular stances in the face of approaching elections.

The President will be even more worried by the increasing pressure at home. Four bills mandating action on global warming have been introduced in the first few days of the new Congress, and more are to follow as it becomes one of the hottest issues on the Hill. A growing number of state governments, including Republican ones such as Arnold Schwarzenegger's in California, are taking increasingly tough measures of their own.

Last week a powerful group of evangelical Christian leaders, one of the President's key constituencies, issued an "urgent call to action". And tomorrow - in the most significant move of all - top business leaders headed by Jeffrey Immelt, the General Electric boss, will choose the day before the State of the Union address to call for cuts of up to 30 per cent in carbon dioxide emissions.

In the past the Prime Minister has refused repeated pleas to attempt to mobilise US opinion against the President's obscurantist position. Now, by contrast, he appears to be pushing the boundaries of acceptable interference in another country's internal affairs on the issue. Last month he met both Congressional and business leaders, and last year he formed a public alliance with the man whom Californian officials now call "the greenhouse gas Terminator".

Mr Bush is beginning to show the first signs of movement, publicly linking climate change with his long-standing concern about increasing US dependence on energy imports.

Both will be addressed in Tuesday's speech, in which the President will announce a vast expansion of bio-fuels and new measures to encourage renewable sources and energy efficiency. He is also considering limited measures to reduce greenhouse gas emissions from power stations, but - despite optimistic reports - is unlikely to signal a full global warming U-turn.

Mr Blair and his allies hope that this will come in Heiligendamm, with the world's most powerful leaders agreeing to set a limit on the amount of pollution the climate can tolerate, leading to national allowances that can be bought and sold in international trade, and help for developing countries.

Whether Mr Bush will agree to this is anyone's guess; he remains the most hawkish member of his administration on the issue. But if Mr Blair can pull it off this last, and perhaps greatest, throw of his premiership, it would be churlish - despite everything - to deny him his desperate desire to go out on a high.