Mr Holbrooke said that he had had a "very good talk", in the midst of "an intense phase of shuttle diplomacy". Nobody could describe that last phrase as an exaggeration: Mr Holbrooke flew from Sarajevo to Belgrade on Saturday, held talks with Mr Milosevic till midnight, then left yesterday morning for the Croatian capital, Zagreb. From there, he went briefly to Sofia to brief Bulgarian leaders, and was due to return to Sarajevo today.
Locals are impressed by his energy, if nothing else: the Belgrade weekly Nin devoted a column to explaining the history of shatl-diplomatia, "an American speciality at times of great urgency". Earlier in the weekend, Mr Holbrooke had warned against "premature" talk of a ceasefire. He argued: "This isn't an express train, and never was."
To an extent, Mr Holbrooke's caution may stem from the desire to make the final moment - when he pulls the rabbits out of the hat - all the more dramatic. The cover headline in Vreme news magazine describes last week's deal, brokered by Mr Holbrooke, as "the New York Jajce" - a reference to the historic meeting which marked the founding of post-war Yugoslavia. Vreme speculated on a historic scene that would include the three presidents - Mr Milosevic, Franjo Tudjman of Croatia and Alija Izetbegovic of Bosnia - standing with President Bill Clinton on the White House lawn, to sign a peace plan. "This scene would have been unthinkable until now, but after the performance which Holbrooke directed in New York, it seems that we may see this movie soon."
There are, however, good reasons to take Mr Holbrooke's caution at face value. Indeed, it may be understated. Vreme concluded: "It's a very different matter, how things will work out in practice. It is unclear whether these questions keep the Americans awake at night. To paraphrase Churchill: 'They won't have to live in Bosnia after the war.' "
Questions of territory, including the future of Sarajevo and the proposed corridor to the Muslim-held town of Gorazde, will be hard to settle. But these problems pale into insignificance by comparison with the long-term difficulties of sustaining a deal, however the borders are drawn.
The New York deal envisages a Bosnia consisting of two equal entities - the Muslim-Croat federation on the one hand, and the Republika Srpska on the other. Within the Muslim-Croat federation, however, the Croats now offer little loyalty to a Bosnian government and look almost entirely to Zagreb.
The Bosnian Croats have Croat currency, a Croat army, and the Croat flag. Mr Tudjman has never hidden his desire to carve up Bosnia; recently he noted that Zagreb accepted the federation only "for strategic reasons". President Tudjman has talked too, in an interview with Le Figaro, of the need to "Europeanise Bosnian Muslims so that they can be integrated in European civilisation" - this from a man who allows fascists to run riot and who has given his blessing to "ethnic cleansing" in recent weeks. If the Croats do not sabotage their federation with the Muslims, it will be a miracle.
On the Serbian side, things look equally bleak. The very name of the Serbian part of Bosnia shows how bizarre things have become. There will be two adjoining republics: the Republika Srpska, or Serbian Republic (in Bosnia), and the Republika Srbija, or Republic of Serbia. Western journalists use the original Srpska title for Radovan Karadzic's fiefdom and the English word Serbia for Mr Milosevic's country in a desperate attempt to help the reader. But the reality is that the two names are separated only by a frightening piece of syntax. It seems likely that the adjectival Srpska Republic will in due course seek to merge with the big-brother Srbija.
Vetoes are built into the New York deal to prevent such secession, which brings us back to where we were in spring 1992, before the war broke out. The Croat-Muslim majority, unwilling to let Bosnia come under the thumb of Mr Milosevic, voted for the independence of a multi-ethnic Bosnia; the well-armed Bosnian Serbs wanted to break away, to be with Serbia.
A few years ago, the incongruity of the present deal might not have mattered. "Bosnian" was a real concept for most Yugoslavs - as easy to grasp as "American", when talking about ethnic Italians, Irish or Poles. Bosnian Serbs were as conscious of their "Bosnian" identity as they were of their Serb identity. That is long gone. For the deal to work out, tolerance on all sides is needed. But the politicians - Messrs Milosevic, Karadzic, Tudjman - have destroyed the possibility of tolerance in the past three years.