United States President Barack Obama labelled it a "moment of opportunity"; Secretary of State Hillary Clinton said on Wednesday at the Council on Foreign Relations in Washington, "Both sides and both leaders recognise there may not ever be another chance."
They were referring to the direct talks between Palestinians and Israelis, the first for two years, that had begun a week earlier in the US capital and which are aimed at securing a peace settlement in the long-running conflict between the two peoples.
The talks will resume for a second round in Egypt next week.
To date, precious little has emerged from the encounter which, among others, brought the central players, Israeli Prime Minister Benjamin Netanyahu and Palestinian President Mahmoud Abbas, to the negotiating table.
This in itself was a small victory in the long - seemingly endless - march towards reconciliation and an amenable two-state solution, the ultimate goal of the talks.
And given the historic and repeated fallibility of such negotiations, and the weight of expectation and commitment devoted to them by successive US administrations, Mr Obama and Mrs Clinton could perhaps be forgiven for taking a deep draught in this particular last chance saloon.
Whether such optimism, cautious or otherwise, is justified, is another matter entirely.
As if to underscore this, the talks began against a backdrop of further bloodshed, when Hamas shot dead four Israeli settlers on the West Bank.
The episode was a chilling punctuation of one of the central agenda issues in negotiations: construction of settlements by Israeli settlers on the West Bank which, in turn, goes to the larger matter of the border parameters of any future Palestinian state.
The killings threatened to end a temporary freeze on construction due to culminate in a couple of weeks.
Chief Palestinian negotiator Saeb Erekat is on record as saying his side will walk out of the talks unless the freeze is made permanent for the West Bank, and applied to East Jerusalem which does not come under its terms.
But mindful of the hardline pro-settlement hawks to the right in his coalition Government, Mr Netanyahu will not easily make such promises.
So in this matter alone there are mountains to climb.
Further to the problematic "border" matters and formalisation of a Palestinian state following current contours of the West Bank would, de facto, surely bring an end to the settlement question the final status issues of the talks include: Jerusalem and how its future fits into any solution; and the fate of the Palestinians who lost their homes and land in the conflicts of 1948 and 1967.
The flip side of the latter is the demand by Mr Netanyahu that the Palestinians recognise the Jewish nature of the Israeli state.
Given that it is an article of faith by Hamas radicals to deny this, Mr Abbas must walk his talk with accusations of "collaboration" and "betrayal" from some of his own people ringing in his ears.
He will not be the first Palestinian leader to find himself thus confronted.
In the end, even that great bogeyman of Palestinian liberation, Yasser Arafat, was insufficiently extreme for the young hotbloods of Hamas.
Nor is it Mr Abbas's curse alone.
Mr Netanyahu will be negotiating with the ransom notes of his coalition's hardliners in his breast pocket.
What is in it for him?History, perhaps? Despite the West Bank killings and other isolated outburst of violence, and as the country puts behind it the public relations mire of the Gaza aid convoy incident, Israel is enjoying a relative period of calm.
Its economy, largely unconstrained by the banking excesses and financial shenanigans that have plagued the United States and much of Europe, is growing.
If a leader could regularise, even make permanent such stability by dealing to the "Palestinian problem", a place in history would be secured, along with the support of much of the world in confronting the country's biggest long-term threat: Iran.
But the moderation and spirit of compromise this would require is in short supply in the Middle East, which is why Mr Obama and Mrs Clinton are making such encouraging noises and putting their full weight behind the talks.
They know, as do the other leading participants, that failure will write in blood the foreseeable future of Israelis and Palestinians, and do nothing to lessen the ascendancy of reckless radicals across the entire region whose rhetoric constantly pushes for confrontation and war.
