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Despite the full-court press, many Republicans remain unconvinced. On CBS’ “Face the Nation,” Mr. Graham said: “If you want to have a chance of passing START, you’d better start over and do it in the next Congress, because this lame duck has been poisoned.”

Mr. Graham was referring to the approval of a stand-alone bill to repeal the military’s “don’t ask, don’t tell” policy that banned openly gay service members.

Mr. Graham’s opposition to the treaty is significant because he was among the senators who voted to take up ratification during the lame-duck session, breaking with more conservative Republicans and suggesting he could be persuaded to vote for ratification.

Democrats on Sunday were nonetheless optimistic. Sen. Richard J. Durbin, Illinois Democrat, said Senate Democrats have enough votes during the lame-duck session to ratify New START.

“I think we do” have the 67 votes, or two-thirds majority, Mr. Durbin said on “Fox News Sunday.” “We’re moving forward.”

Mr. Biden later backed that claim on NBC’s “Meet the Press,” stating, “I believe we do” have the votes for approval.

The Senate debated the treaty in between other business on Saturday and Sunday.

Senate Minority Whip Jon Kyl, Arizona Republican, had offered Mr. Reid a deal this month that called for the Senate to take up debate on New START in late January and to hold a vote on treaty ratification Feb. 5. Mr. McCain disclosed the outlines of the deal on Friday in a floor speech.

Congress has passed major legislation during the lame-duck session, including an extension of tax cuts for all Americans and allowing gays to serve openly in the military.

However, Republicans say they don’t have enough time before Congress adjourns for Christmas to debate or propose amendments to the New START agreement, which treaty proponents say is needed for monitoring Russia’s nuclear forces.

In this Congress, Democrats need nine Republican votes to get the 67-vote two-thirds majority needed to ratify New START. In the next session, Democrats would need 14 Republicans to ratify the treaty.

Mr. Kyl, who said Sunday that he now opposes the treaty, said Mr. Obama’s letter on missile defenses has assured lawmakers of the U.S. commitment to building long-range missile defenses in Europe. However, he said that promise should be in the treaty.

“Tell that to the Russians,” he said. “This treaty needs to be fixed.”

A key swing vote in the debate is Mr. McCain. The senator has not said whether he will support or oppose New START. He has focused on his concerns that the Russians will use the treaty to pressure the United States to limit missile-defense plans.

Mr. Kyl summed up the Republican opposition to New START on Sunday when he said, “I don’t know whether it was because of a lack of direction from the commander in chief or poor negotiation, but one way or another we got snookered.”

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