Only hours before, she had telephoned Yasser Arafat to plead her excuses. She could not come to see him as agreed, she said. She was simply too tired to drive over to Claridges for their meeting. Arafat burst into laughter when the call was over.
Never mind that his own state of health - shocking to behold when only a few feet from him - was far worse than Mrs Albright's. But when it came to Benjamin Netanyahu, a few hours later, Mrs Albright was off in her limousine to meet the Israeli Prime Minister at his hotel.
And what came over most strongly yesterday was Mrs Albright's fear of Mr Netanyahu, indeed perhaps her fear of Israel. Mr Arafat and the Palestine Liberation Organisation had already accepted America's conditions for the 11 May invitation to meet President Bill Clinton in Washington. Mr Netanyahu had not responded. He was flying back to Israel to consult his cabinet. But when Mrs Albright talked to us later - hesitant and sometimes confusing or forgetting questions - she was all praise for the Israeli Prime Minister who is forging ahead with Jewish settlements on the land Mr Arafat wants as part of his Palestinian state.
Mr Netanyahu, we heard, was encouraging. He had produced "new ideas". He was enthusiastic. He was "helpful." She was very grateful to Mr Netanyahu. As for Israel's security demands - which now include a decrease in the number of Mr Arafat's policemen - "it is obviously up to Israel to decide what its security demands are ..."
But that was the whole point. Since Israel, on "security" grounds, is still refusing to give up more than 9 per cent of occupied land - an odd 11 per cent figure surfaced during the day although the Israelis would not officially confirm it - this effectively gave Israel the right to decide on the size of its withdrawals.
When we asked Mrs Albright what all those new ideas were - what possible progress she could be talking about after two days in which Mr Netanyahu and Mr Arafat couldn't even bring themselves to talk to each other on the telephone - we were informed that "more details do not help us to move forward". It was an odd phrase - but not as surprising as the admission that US proposals for Washington talks included an immediate move to what in the Oslo agreement are called "final status talks" - something that Mr Netanyahu has been demanding for the past 12 months.
So what did this mean? According to Mrs Albright, an "accelerated peace process". But a glance at the Oslo treaty shows that it would probably allow Israel to stall on any further withdrawals - or reduce the amount of occupied Arab land it was prepared to evacuate to a mere 25 per cent, if that. And Mr Arafat is not going to get a Palestinian state on this little rump of territory, carved up as it already is by roads exclusively for the use of Jewish settlers.
Yet still Mrs Albright talked of progress - she used the word at least 18 times in just a few minutes. And so did Tony Blair an hour earlier. Only Mr Arafat, partly stooped as he stood outside 10 Downing Street, gave any clue to the fantasy world in which the negotiators were immersing themselves. He had "heard" from Mrs Albright, he said, that there had been "some progress" and he would go wherever necessary to save peace.
It was when I asked him if he did not now regret signing the Oslo agreement with Israel that the old man's eyes suddenly widened and his voice took on its old strength. "The peace agreement I signed was the peace of the brave," he replied. "I signed with my partners Yitzhak Rabin, who paid with his life for this peace. It is our firm duty that we continue with the just endeavour we signed with Mr Rabin and Peres."
There was deliberately no mention of Mr Netanyahu. Indeed, in none of the sound-bites he uttered yesterday did the Israeli leader come close to Mr Arafat's albeit familiar promise. Nor did Mrs Albright. She remarked of America's peace-making efforts that "it's up to the parties [to decide] as to whether we are serving the vegetables well." Perhaps that will be written on Oslo's tombstone. By contrast, Mr Arafat was momentarily in Jefferson mode.
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