The low-key nature of the public proceedings was appropriate. We have been here, or somewhere like it, before; Palestinians and Israelis remember only too well the surge of optimism after the 1993 Oslo accord, followed by the realisation that this was a false dawn.
It is not to underestimate the importance of both sides calling a halt to four bloody years of a conflict that has cost more than 4,500 lives, to recognise that in the sweep of history this is just one, painfully reversible step down a long and obstacle-strewn path to a just and lasting peace.
If Mr Abbas seemed more comfortable in his skin yesterday than Mr Sharon it was understandable. He has scored a considerable success, one that may increase his standing among Palestinians as a man who can influence events.
True, there was no joint declaration, but Mr Sharon had been reluctant to make any formal, public announcement of a halt to military action. Whether his decision to do so had anything to do with this week's visit by Condoleezza Rice - Israeli officials suggest it didn't - his statement was less grudging and equivocal than the advance briefing had suggested it would be.
Mr Abbas has got further and faster than most expected after his election on 9 January. If Mr Sharon's plan to withdraw settlers from Gaza goes ahead on time - and if the prospect he held out of co-ordinating it with the Palestinians is realised - then Mr Abbas cannot fail to reap the benefit. There will be international money and help for Gaza's ravaged economy. And though it was conceived long before Arafat's death as a unilateral step, the first ever reverse of the relentless settlement growth in occupied Palestinian territories will have happened on Mr Abbas's watch.
The dispatch of US General William Ward as a "security co-ordinator" may help too. Mr Abbas does not need ominous muttering from Hamas to remind of him of the fragility of truces. But General Ward's presence gives the US an important stake in this ceasefire as well as acting as a reminder to Israel that the US wants Mr Abbas to succeed.
Mr Abbas, facing internal political challenges, needs more than that. First, he needs the "confidence building measures" Mr Sharon spoke of to have an impact on the day-to-day lives of Palestinians. That may mean Israeli relenting on its determination to release some long-serving prisoners "with blood on their hands". It will certainly mean the dismantling of checkpoints and closures, which bring economic and social misery. But equally it will mean the maintenance of political momentum. And here there was little encouragement for Mr Abbas this week.
Based on its inclusion in phase one of the road map, the Palestinians see the issue of "dismantling the terrorist infrastructure" and disarming the factions as linked to and facilitated by a political process. The more the Palestinian state begins to take shape, the easier it will be to fulfil this. The subtext of Mr Sharon's approach is that it is a precondition of the road map process starting.
The contradiction in the US position appears to be this: Ms Rice and President Bush genuinely want Mr Abbas, and the road map, to succeed. Whether in a Freudian slip or not, Ms Rice seemed to contrast the positions of Mr Sharon and Mr Abbas when she said on Monday "we know" that the Palestinians accept the road map and that Mr Sharon "has said" that he does so.
But at the same time the US also shows little sign of wanting to undermine the belief of many around Mr Sharon that "final status" issues can be put off - perhaps for years. That could deprive Mr Abbas of the momentum he needs.
But if Mr Abbas is allowed to fail, it will puncture a central tenet of the Israeli government over the past four years. For then it would appear that it was not Yasser Arafat who had been the problem after all.