Just a month ago, prospects for the treaty appeared to be bleak, when Mr. Kyl declared that there was not enough time to approve the treaty before the end of the year. Mr. Obama decided to wage a high-profile campaign for the treaty over Mr. Kyl’s objection, risking a large share of his prestige and testing his clout in the new political environment.

To bypass the hostile leaders and win over other Republican Senators, Mr. Obama made a commitment to spend $85 billion over 10 years to modernize the nation’s nuclear weapons complex, so that the smaller arsenal would still be well-maintained and effective. He also gave repeated assurances that he would follow through on development of missile defense in Europe, despite Russian resistance.

The final vote on the treaty came after the Senate disposed of a raft of Republican-proposed amendments Tuesday night and Wednesday morning. Most of them were rejected entirely, but Senator Kerry accepted a few of them as side statements, which do not formally become part of the treaty and therefore do not require renegotiation with Russia.

Among those accepted on Wednesday was one by Mr. Kyl on modernization and one by Mr. Corker on missile defense. Senator John McCain, Republican of Arizona, who had been trying to work out his own side statement on missile defense, joined in backing Mr. Corker’s amendment instead.

Mr. Kerry also accepted on Tuesday night a declaration that the United States should open new talks with Russia within a year to negotiate a new treaty curbing tactical nuclear weapons, the smaller battlefield bombs that are not covered by New Start or any previous Russian-American treaty.

Russia has far more such weapons than the United States, and according to American officials, as recently as last spring Russia moved some of them closer to its borders with NATO nations, as a n response to American missile-defense deployments. Some experts consider these smaller bombs a greater risk of theft or black-market diversion to rogue states or terrorist groups.

The New Start treaty lowers the ceiling on strategic weapons set by previous Russian-American treaties, which it will now supplant. Under the Treaty of Moscow, signed by President George W. Bush in 2002, each side was allowed no more than 2,200 strategic warheads as of 2012. Under the original Strategic Arms Reduction Treaty, or Start, signed by the first President Bush in 1991, each side was required to reduce launchers to 1,600 before the treaty expired last year.