It is a sign of the despair of the past four years of the intifida that both Israelis and Palestinians liked to tell the following joke. Yasser Arafat (or Ariel Sharon) asks God: "Lord, will there ever be peace in the Middle East?" God answers: "Yes, of course, but not in my lifetime."

Watching the white-haired Ariel Sharon and Mahmoud Abbas sitting at the table at Sharm el-Sheikh on Tuesday, pledging to stop killing each other and to seek peace based on the creation of a Palestinian state alongside Israel, may appear to be something of a miracle - a parting of the ocean of blood between Arab and Jew on the shores of the Red Sea. Caution is in order, however.

There have been many false dawns in the Holy Land. I have watched several peace summits in Sharm el-Sheikh that have ended in failure. It was less than two years ago that Abbas and Sharon met at the other end of the Sinai, in Aqaba. Their speeches then were almost identical to their lofty words of peace this week.

Then, as now, Palestinian militants declared a truce. Then, as now, Abbas pledged to end violence and promised to build a democratic Palestinian Authority that would have a monopoly on weapons. But the truce collapsed, the suicide bombers started blowing themselves up once more, Israel halted the political process and Abbas resigned in frustration.

Why should it be any different now? The main reason for hope is that Arafat is history and Sharon wants to make history. Both sides are tired of a conflict that has killed more than 4000 people and set back the cause of peace by more than four years.

The flurry of recent moves towards peace - Israeli withdrawal from West Bank cities, the release of Palestinian prisoners, the imminent return of Egyptian and Jordanian ambassadors to Tel Aviv, and the return of the Bush Administration to active diplomacy in the Middle East - are all the product of Arafat's death.

The main reason for hope is that Arafat is history, and Sharon wants to make history."Abbas won a convincing election victory. He now has a popular mandate to end the Palestinian uprising and no longer has to look over his shoulder at Arafat who, as they say in the Middle East, "spoke out of both sides of his mouth" on the question of violence.

Sharon is also a changed man. Since the ill-fated Aqaba summit, he has caused an earthquake in Israeli politics by forcing through a "unilateral" withdrawal from the Gaza Strip, removing all soldiers and settlers from the territory by next northern summer.

He has brought Shimon Peres' Labour Party back into his coalition. Sharon, the political "father of the settlements", is now planning to use the army to evict thousands of settlers from Gaza, under fire if necessary. The man known by Arabs as the "Butcher of Sabra and Chatila" (because of the massacre of Palestinians during the Israeli invasion of Lebanon that Sharon masterminded as defence minister in 1982) is now embraced by Egypt as the only Israeli statesman who can bring peace.

The death of Arafat and the death of Sharon's dream of "Greater Israel" have removed two large obstacles to peace-making in the Holy Land. But the way to peace could still be blocked by extremists, whether they be Islamic militants seeking a return to the glory of the caliphate or Jewish hardliners convinced that they are bringing about the coming of the Messiah.

Is the ceasefire agreed by Hamas and Islamic Jihad more than just another brief respite? Will settlers plan some new provocation to disrupt the negotiations, as they did with the 1974 disengagement accords, the 1979 Camp David accords and the 1993 Oslo accords?

Will Abbas have the guts to risk a Palestinian civil war to stop militants if they resume attacks? And if he does, will Sharon have the guts to hold back from retaliating? These are just a few of the questions that will be asked in the coming weeks.

There are even bigger uncertainties. Sharon and Abbas seem able to agree on the first steps: ceasefire, normalisation of Palestinian life, Israeli withdrawal from the Gaza Strip, creation of a Palestinian mini-state.

Then what? Sharon is giving up Gaza, the part of the occupied territory that Israel finds most difficult to hold on to; but he says nothing about the West Bank, the part that Israel finds hardest to give up. The most difficult issues - the borders of the Palestinian state, the fate of settlements, the status of Jerusalem and the return of Palestinian refugees - are being left to last, like so many ticking bombs.

The international "road map" for peace offers no precise destination. It sets out in great detail the first milestones, but says almost nothing about how it will "end the occupation that began in 1967". While the Palestinians are taking the first steps in the road map by promising to end violence, Israel insists that its own initial obligations - the dismantling of illegal settlement outposts and freezing all settlement building - will be postponed to an indefinite date after the Gaza withdrawal.

For Israelis and Palestinians, "peace" means very different things: Israelis want "security" and an end to conflict, while Palestinians want "justice" and independence. Unless Sharon and Abbas can agree, in secret if necessary, on the kind of final peace they seek to achieve, then any interim steps they take will prove short-lived.

The outside world can help. It can assuage Israel's security fears by providing military observers to ensure that Gaza does not become a terrorist outpost, and it can help Palestinians by offering money and expertise to rebuild their institutions.

George Bush can help both sides by spelling out more of the detail of a permanent peace agreement. He has already backed Sharon by declaring that Israel can retain blocks of settlements in the West Bank and that there will be no "right of return" for Palestinian refugees to Israel. He now needs to support Abbas. He should say that Palestinians have a right to the West Bank, territorial compensation for any land annexed by Israel and a share of Jerusalem, including the Old City.

This is all very similar to the "Clinton parameters" issued in December 2000. Call it the "Bush vision" if need be, but the US President needs to talk about freedom for Palestinians with the same strength as he talks about freedom for Iraqis.

Then maybe, just maybe, God might smile on the Holy Land and bring peace in our lifetime.

Anton La Guardia is diplomatic editor of The Daily Telegraph, London, and author of Holy Land, Unholy War: Israelis and Palestinians.