At Dulles airport in Washington, Gordon Brown is waiting in the queue to clear customs like everyone else. He waits patiently while an official checks his passport. She looks up briefly; there is no flicker of recognition. He may be on course to be Britain’s next prime minister but there are no special privileges, no red carpet — not yet.

Brown remains composed until one of his aides wanders off down the wrong immigration channel and gets ushered away by a burly customs guard wearing rubber gloves. “They think he’s got something in his suitcase that he shouldn’t have, do they? It’s only my speech to Labour conference,” he quips.

The chancellor has reason to be cheerful. Coming to Washington to chair the annual meeting of the International Monetary Fund (IMF) is one of his favourite tasks, but it is not the world’s money markets that have put a spring in his step.

After a third election victory, Blair is at last preparing to leave Downing Street and for Brown the Labour leadership and the keys to No 10 are all but within reach.

Even for a politician as well rehearsed as this, Brown finds it difficult to disguise his ambition; he exudes the persona of a man who, having waited in line behind Blair for more than a decade, only needs his passport stamping before walking into No 10.

“What I want to do is deepen and broaden our appeal,” he said at one point, before quickly correcting the “I” to “we”.

Brown agreed to talk to The Sunday Times on his way to the IMF meeting in Washington, but it is this week’s Labour party conference in Brighton that is the main event. It is the first conference since Blair announced his intention to stand down and it marks the first hurdle in the final straight on Brown’s marathon race to become PM.

All are agreed that the going looks good for the chancellor. On Friday Tessa Jowell, the ultimate Blairite cabinet minister, pretty much anointed him as Blair’s successor, predicting that the chancellor would carry the torch for new Labour’s free market reforms.

“He has subscribed to the policies that have built this progressive consensus with the same degree of enthusiasm as the rest of the government,” she said.

Even Charles Clarke, one of Brown’s arch-rivals, acknowledged that it was “very likely” that Brown would become prime minister and that “would be a good thing”.

BROWN also has the backing of the left, including the unions, which see him as a man after their own heart — not quite as an enemy of capitalism, perhaps, but as a social engineer and redistributor, a man who understands the needs of the working class.

Only the likes of Alan Milburn, the über-Blairite former minister for health with whom Brown clashed over hospital autonomy, pose a threat. The self-appointed “outriders” for Blair’s legacy suspect that under Brown the clock could be turned back, the middle classes lost and power eventually handed back to the Tories.

Milburn will this week send a warning shot along those lines, arguing in Brighton that the “ideological battle” has yet to be won. His prescription will be “more, not less, new Labour”.

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