Defying the logic of the conflict, the bleak pronouncements of the key US negotiator and the new cloud cast over the "peace process" by Thursday's war crimes indictment of Mr Milosevic and four of his closest allies, the former Russian prime minister pronounced himself "very satisfied" with his latest talks with the Yugoslav President on Friday evening.
Undeterred by Friday's pointed reminder from Strobe Talbott, the US Deputy Secretary of State, that he was in Belgrade as a representative of Russia and Russia alone, Mr Chernomyrdin yesterday reported back to his Foreign Minister, Igor Ivanov, and sought to launch a new round of contacts with the West.
Thus the Kosovo war, now into its third month, approaches its first diplomatic moment of truth. If flesh can be put on the bones of what Mr Chernomyrdin claims is an understanding with Mr Milosevic, then his next trip to Belgrade - possibly on Wednesday or Thursday - could be in the company of Martti Ahtisaari, the Finnish President, the West's designated representative in its dealings with Yugoslavia, whose presence would be a sign that a real breakthrough is at hand.
But the devil, as always over Kosovo, lies in the detail. According to Moscow, the Yugoslav leader is now ready to go along with the outline proposals hammered out by the Group of Eight foreign ministers a month ago, providing for an "effective" international peacekeeping force, and the withdrawal of Serbian troops and police from the province.
But last night Nato had no details of new concessions from Mr Milosevic, who the alliance insists must meet its key demands: a Nato-dominated and heavily armed peacekeeping force, withdrawal of "all" Serbian troops and paramilitary units, and turning Kosovo into an international protectorate.
Until then, Nato says, the bombing will go on. And in a signal that Moscow may be preparing itself for failure, Mr Ivanov yesterday accused "leading Nato countries" (ie, Britain and the US) of "not fully understanding" Russia's efforts. If the West is unimpressed by the outcome of Mr Chernomyrdin's latest mission, then he would have little choice but to carry out his threat to abandon Russia's peace mission, pinning the blame - as he did in an extraordinarily hostile comment in the Washington Post - squarely on Nato.
So what happens next? For all the West's ritual expressions of yearning for a diplomatic solution, the prospects of one soon are non-existent. If Mr Chernomyrdin now does what he has been threatening to do for a week, Russia will abandon its peacemaking efforts as a waste of time, and use its veto to block whatever the West suggests at the United Nations.
True, there are other possible go-betweens. The UN Secretary-General Kofi Annan is one. Greece, the Nato member most sympathetic to Mr Milosevic, is another; it opposes the bombing, and its former prime minister, Constantine Mitsotakis, is a good friend of the Yugoslav President. But if Russia cannot get him to change his tune, there is little likelihood anyone else can.
So the bombing will grow even fiercer. On Friday, Nato flew a record 792 sorties in one day: with summer's clearer weather, and extra aircraft in Hungary and Turkey, that will increase. Resolute diplomacy last week by the Foreign Secretary, Robin Cook, ensured that talk of a bombing pause "to give Milosevic a chance" has vanished.
But if the allies hope the Belgrade regime will buckle, or public dissent will lead to an internal uprising, there is little sign of either. The indictment reduces what incentive there was for Mr Milosevic to reach a settlement. Indeed, Nato leaders are signalling that none is possible while he is in power. But he has no obvious rival, and a successor is likely to be more hardline.
If there is no submission, Nato will have to win militarily, probably on the ground. It is slightly improbable, perhaps, to compare the hard hills and deep valleys of a remote Balkan province with a whitewashed Spanish mission church in flat and dusty Texas.
But with every passing day, Kosovo more resembles Mr Milosevic's Alamo, to be defended to the death.