An account of this scene is one of the highlights of the first episode of Israel and the Arabs: Elusive Peace; and it's a good illustration of what it is that makes Norma Percy's brand of documentary-making so compelling and so widely admired. In her hands - in series such as The Second Russian Revolution (1991), The Death of Yugoslavia (1995) and Endgame in Ireland (2001) - the dry, abstract world of international politics has been given a human face, and turned into thrilling, and occasionally comic, drama.
It's an approach that has brought her and her colleague Brian Lapping plaudits from press around the world ("the Rolls-Royce of documentary-making", said the Wall Street Journal), and just about every major TV award going. Through a combination of archive footage and interviews with presidents, prime ministers, foreign ministers and civil servants - everybody who matters - you are given the impression of sitting in on history. You can get a sense of the way clashes of personalities and, sometimes, unexpected sympathies between enemies, can shape our world.
With Elusive Peace, though, the process was not easy. The series, a sequel to The Fifty Years War (1998), follows the agonising negotiations between Israel and the Palestinians since 1999, as a shaky truce has spiralled into violence. The spark for this series was a remark Bill Clinton made when Percy interviewed him in 2001 for Endgame in Ireland, her series about the Northern Irish peace process. Recalling the peace talks held at Camp David in July 2000, Clinton sighed over Yasser Arafat's refusal of an Israeli proposal to share Jerusalem: "What a deal he had on offer."
In theory, Percy and her team had an advantage, already knowing the background and the major players. But as she told me at the offices of Brook Lapping in Kentish Town, London, last week, this time a year ago she was sitting on a terrace in Jerusalem with a glass of wine wondering whether to call the whole thing off. She recalls saying, "We don't have anyone." And as she says, "What they hire us for is to get everybody."
The central difficulty was the whole business wasn't concluded and a lot was still at stake; and several of the protagonists had good personal reasons for not finding time. Barak, ousted from office in 2001, was trying to relaunch his political career. Clinton was touring the world, promoting first his autobiography, and then tsunami relief. Arafat was the most complicated of all - his constant terror of assassination meant that even to get in his front door involved a number of trips to Ramallah, waiting in hotel rooms for a phone-call that might or might not come. When they did finally meet, over lunch in his compound, the conversation was stilted and awkward, until Mark Anderson, one of Percy's directors, asked Arafat who was the most memorable of all the great men he had met: ice broken, he wouldn't stop talking.
Why does Percy think the top people do talk to her? The simple answer is, "They want to get their part in history": Percy's programmes are a way of getting their views on record. But though she doesn't conduct all the interviews herself, I suspect her personality is an important factor. I've known Percy for five or six years, since she wrote to thank me, but also - characteristically - chide me for an admiring review I wrote of Playing the China Card (1999), a two-part film about America's complicated relationship with China.
She is tiny and, at first sight, almost worryingly frail - an impression heightened by the way she hunches over, as if bowed down by trouble; she speaks quietly, in a New York accent with the edges rubbed over by more than 30 years in London, and has a devastatingly self-deprecating sense of humour. She says herself that she thinks one reason politicians will talk to her is that is so unthreatening. The truth is, she is buzzing with nervous energy, and not so much hunched as wired tight. I am pretty sure that I would not want to work for her, knowing as I do the sort of commitment she takes for granted - happily working late into the night, for weeks on end. On the other hand, she is very loyal to her colleagues. Several times during our conversation, and in an e-mail afterwards, she insisted that I underline the contribution put in by her directors, Mark Anderson and Dan Edge.
Another factor in her success is the financing, which comes mainly from the BBC but also from a coalition of overseas broadcasters, including Arte in Europe and PBS in the US. Co-financing doesn't always bring rewards: Horizon, BBC2's flagship science programme, has been ruined by American money, which has contributed to a creeping Americanisation of subject matter, and a simplistic, over-dramatised style.
For Percy, though, the promise of a wide audience - and particularly an American audience - is essential as bait for her interviewees. She admits that co-production money does mean that: "We tend to look for stories with an American angle. But then, big international political stories do tend to have an American angle."
The generous budgets she usually secures mean, too, that she has more time to do research: "It takes time to even find out that these meetings happen," she says. And a lot of her interviewees say no to the first request - because they have a country to run, or because the issues being discussed are too sensitive, or because they don't know Percy and haven't yet realised that it's going to be simpler just to do what she asks.
I wonder about her answer to the Arafat question: who has impressed her most? "Most of them," she says, "you come out liking more than when you went in."
Her premise is "that politicians are serious people grappling with serious problems" (though she makes an exception for Slobodan Milosevic). They are also - Milosevic aside - honest with her; at any rate, she says it is rare for their stories to contradict one another.
If she had to pick one, though, "Clinton bowled me over". "The amazing ability to make you feel like the only person in the room", of course; but also, from an interviewer's perspective, a depth of recall, and an ability to understand other people's views, to assess their bottom lines. "You can see what makes him such a good negotiator." This comes across in Elusive Peace, and gives it a tragic edge: if Clinton couldn't nudge the Middle East into peace, you wonder who on earth can.
'Israel and the Arabs: Elusive Peace' starts on Monday 10 October, 8.30pm, BBC2
'Elusive Peace: How the Holy Land Defeated America' by Ahron Bregman is published by Penguin, £8.99