"Russia will participate in these operations," Mr Yeltsin said after the three-hour session. But the exact form of that participation will only be decided - if it can be decided at all - by a meeting of their defence ministers "within a week".
For his part, Mr Clinton reported "some progress" with Mr Yeltsin on the nature of the peace-keeping force. But, he added, "the more we say, the worse it will be," indirectly highlighting the delicacy of the task facing William Perry, the US Defense Secretary, and the Russian Defence Minister, Pavel Grachev. While the US wanted Russian involvement, "neither side is giving up what it holds most important".
For their third meeting of the year, the two leaders chose the deliberately symbolic site of Hyde Park, the family home 50 miles north of New York City of Franklin Roosevelt, under whose presidency the US and the Soviet Union were allies against Nazi Germany from 1941 to 1945.
But despite the outward warmth of the occasion, complete with an exchange of welcoming bearhugs and the laying of red roses at FDR's grave, there was no hiding the differences between Moscow and Washington, as wide as any in the four years since the demise of the Soviet Union. Mr Yeltsin did not soften his hostility to Nato's plans to expand to the east, while the US failed once more to make Russia think again on the $1bn sale of two nuclear reactors to Iran.
Although the two Presidents said they would seek a complete nuclear test ban agreement next year, and speedy ratification of the Salt-2 strategic arms treaty, US officials denied Mr Yeltsin's claim at the press conference that they had struck a final agreement to update the 1990 CFE treaty on conventional forces in Europe, permitting Russia to keep more equipment close to its southern borders to ward off possible secessionist threats.
In his speech on Sunday at the United Nations, Mr Yeltsin declared that Russia would only take part in the peace-keeping mission that would follow a settlement of the Bosnian war if it were under "a strict mandate of the UN Security Council", where Moscow has a veto.
As far as the US is concerned however, it will not commit troops to the force unless it is under a single and unambiguous Nato military command.
In the longer run, Moscow's objections to the alliance taking in as new members former Warsaw Pact countries in central and eastern Europe may be even more troublesome. Despite assurances - most lately from Tony Lake, Mr Clinton's national security adviser - that Nato expansion would be "gradual and open" so as not to unsettle Russia, Mr Yeltsin is unimpressed.
Making matters worse are the domestic politics of both countries, where presidential elections are due next year, and in which each leader will be under pressure from the right.