Confronted by two women, both of whom claimed ownership of the same baby, Solomon suggested cutting the child in half. At that, one of the women offered to let the other take the child, if only the king would spare its life. Seeing her concern, Solomon knew instantly that she was the true mother.

The problem for Clinton, playing a latter-day Solomon over the fate of Jerusalem, is that neither claimant gives the required response. One would happily split the child in two; one insists on keeping it while allowing the other to look after parts.

The solution - for both Jerusalem and the Occupied Territories - is simple in theory. It means resolving the rival claims of sovereignty and control so that granting ownership to one side is balanced and moderated by granting rights to the other. The difficult part is how to do this fairly.

There is little doubt that a deal can be worked out over the West Bank, creating a Palestinian state while allowing the Israelis to retain some settlements. But the Palestinians are worried at the degree of control this would mean ceding to the Israelis. The question is how to meet the security needs of the settlers and provide access to them from Israel proper without making a mockery of the Palestinian state in which they would be situated.

Jerusalem is more difficult, involving matters of principle, since both sides want it as their capital. The Israeli prime minister, Ehud Barak has proposed extending the city’s boundaries to accommodate a Palestinian seat of government but insists on having sovereignty, even though he might not object to seeing Palestinian flags flying in parts of it.

On the Palestinian side, however, Yasser Arafat cannot afford to concede sovereignty of East Jerusalem to the Israelis. Apart from finishing him politically, it would cause fury in the rest of the Arab and Muslim world: besides the Jewish and Christian interests, Jerusalem is the third holiest city of Islam.

One approach would be to attempt a verbal fudge, but that will probably not work because whatever an eventual agreement says about Jerusalem is bound to be closely scrutinised by critics.

Clinton may therefore have to propose a third way which might, for instance, involve internationalising or "Vaticanising" the old (religious) part of the city. Some Israelis would certainly reject that, but if he can come up with a solution which is acceptable to both sides it will be nothing short of a triumph. All the other failings of his presidency could deservedly be forgiven, if not forgotten.

Despite the news blackout at Camp David, it is clear from leaks to the Arab and Israeli press that the talks have now moved from the first phase - establishing the bottom line on each side - to the stage where the Americans seek to close the gaps.

This is the most delicate part because Clinton not only has to demonstrate to both sides that he has some workable proposals and earnestly wants to strike a deal, but must also secure the confidence of the Palestinians who, up to now, have regarded his administration as more concerned with Barak’s problems than their own.

Apart from ideas, the main things the Americans can bring to the table are money and security guarantees.

Money can lubricate a deal by making minor concessions easier to accept. Clinton will almost certainly make an attractive, "unrepeatable", offer of aid to a Palestinian state, and more money would help towards solving the refugee problem.

Money would also be needed for construction work to provide sensible and secure movement within Jerusalem, between the West Bank and Gaza, and between Israel and any remaining settlements.

The Israelis, in addition, will want money to resettle displaced settlers behind the 1967 borders, and money for weapons - essential, they will say - to guarantee their security.

The cost would certainly run into tens of billions of dollars. Both sides will be looking for firm promises from the US, but Clinton cannot be sure how Congress will respond to what is perceived by some as his personal "legacy project".

After 50 years of conflict it would scarcely be surprising if the talks failed - though barring a dramatic walk-out, a failure would more likely be described officially as an adjournment, if only to hold off Arafat’s threatened unilateral declaration of a Palestinian state.

Whatever happens on other issues, the refugee question cannot be totally resolved at Camp David because it involves other countries who are not represented - notably Jordan, Lebanon and Syria - but a framework for a solution could be established.

The most easily achievable outcome, an incomplete deal creating a Palestinian state in the West Bank and Gaza but postponing the question of Jerusalem, seems the least likely. At this stage, neither side regards such a deal, on its own, as worth having.

The Palestinians remain suspicious of Clinton’s assumed role as an honest broker, and some suggest it would be better to wait for the next president.

Barak’s problem, apart from his enemies at home, is the opposite. Future American administrations can scarcely be more sympathetic to Israel - and probably less so. Israel has already served its strategic Cold War purpose for the US and is now simply one among several American allies in the Middle East. That change has not yet filtered through into American public opinion, but it will do eventually - as Barak is well aware. A deal now, even with some painful concessions, could be a lot better for Israel than a deal later.

Clinton’s incentive is no less than Barak’s, and even more personal. Success now would not only give a huge boost to the Democrats in their election campaign but redeem his presidency for the history books.