There now follows a party political broadcast for the Blair Out Now Party. That was how the Bush-Blair news conference at the White House could have been introduced on British television. Attitudes in this country to this particular special relationship are completely fixed, mainly negative and largely impervious to facts. When the St Petersburg tape revealed that, in private, Tony Blair did indeed have a rather different take on the Lebanon crisis - "you need that done quickly because otherwise it will spiral" - he was still derided as George Bush's poodle.
Denunciations of Blair's closeness to Bush are too often a substitute for debate about the rights and wrongs of the Middle East. Almost the only good argument for Blair to stand down soon is that it might force some of his critics to deal with the implications of their position. Where, for example, does Gordon Brown stand on Hizbollah? It seems unlikely that, as prime minister, Brown would have taken a different view from Blair. The Chancellor takes an interest in the Middle East. He was in Tel Aviv in November. He gets on well with Ehud Olmert, the Israeli prime minister. His main emphasis is usually on economic development as the key to peace, as would be expected of a finance minister. But he is also a pro-American, and it is Blair's closeness to Bush, and through Bush to Israel, that is the focus of much fury.
Brown is different from Blair in two important ways, however. One is that he has to face the voters again. If he becomes prime minister, he needs to win back as many as possible of the anti-war coalition that defected to the Liberal Democrats last time. The other is that he did not order British troops into Iraq, which at least makes it possible for him to make the arguments without being dismissed as a warmonger when he opens his mouth.
Thus, although his basic position would be the same as Blair's, he might have put it very differently. Past experience of how Brown fudged issues as various as the Millennium Dome and the euro suggests that he would call for an immediate ceasefire if five conditions are met. Those conditions would have included Hizbollah freeing the Israeli soldiers and pulling back 20km from the Israeli border, as spelt out by Condoleezza Rice. Brown would also sound more concerned than Blair about the civilian casualties.
The same factors explain the puzzle of the Conservative Party's stance. David Cameron does not support an unconditional ceasefire either, but he has allowed William Hague, the shadow foreign secretary, to describe the Israeli attack on Hizbollah as "disproportionate". This is nonsensical. How does one respond "proportionately" to the deliberate random killing of civilians? But it makes people in this country feel better when faced with terrible television pictures of the collateral damage caused by Israeli attempts to root out Hizbollah.
As President Bush said at his news conference with Blair, "The temptation is to say, it's too tough, let's just try to solve it quickly with something that won't last; let's just get it off the TV screens. But that won't solve the problem." It requires political leadership to engage with public opinion and explain why an unconditional ceasefire is a bad idea. Blair has been perfunctory in doing so. He knows what he thinks; he doesn't have an election to win; history will be his judge.
Brown, on the other hand, would have to engage with the TV-watching, vote-wielding doubters. Brown would therefore be forced to make a better fist than Blair of explaining and selling a British policy that insists on dealing with - rather than avoiding - the problem of Hizbollah.
The crisis in Lebanon exposes a contradiction in Blair. Privately, he expresses his frustration that Brown does not do more to set out his stall as a future prime minister. Whenever Brown tries, though, Blair complains about being undermined. Blair also likes to portray himself as soaking up all the punishment of unpopular but necessary long-term decisions, to leave his successor with a cleared site on which to build his own monuments. According to my well-informed colleague Steve Richards, the Prime Minister predicts that, when he goes, many of the discontents of the past nine years, accumulated under the heading of "trust", will be expunged. But he wants to go on being the Chancellor's human shield for a while yet.
The timing of Blair's departure is complicated by the most awkward contradiction of all. It is that between that part of the Prime Minister (the larger part) that accepts that Brown will succeed him and wants to be graceful about it, and the part that still harbours the hope of an alternative. That mischievous part emerged for a split second a few months ago when Michael Parkinson asked him: "Have you told your next door neighbour when you're leaving?"
Blair replied, "Well, when I move out!" before giving the "serious" answer about their 23-year partnership. That division in Blair's personality is reflected among his advisers. One of them admitted to me that they had not yet "moved to the point of identifying a viable alternative candidate". This ultra-Blairite said: "You'll never be able to out-organise GB, but what you might be able to do is out-mood him." He thought the tide in the Labour Party and the country might be shifting.
It is a bit late for that, though. Blair is left with only two realistic options for his speech to the party conference in Manchester in September. He can say he will stand down as soon as a leadership election is held. Or he can say nothing about his intentions. (He cannot give a "timetable" because he would be instantly powerless; and singing a song, as James Callaghan sang "Waiting at the church" to the TUC in 1978, would be a Bad Idea.)
I suspect, therefore, that we will have to wait a little longer before we discover what Gordon Brown's foreign policy is made of. And whether the ferocious anti-Americanism that has infected public opinion will subside under a new prime minister, who might be prepared to argue with it rather than simply dismiss it.