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“Those tours are governed by the Secret Service, which is not part of this amendment, so that would not be affected,” he said.

Mr. Coburn countered that the tours were, indeed, at stake in the vote.

“The National Park Service does have something to do with White House tours because they can take this money and reallocate it. It is not a Secret Service problem. It is a national park problem,” he said.

The Senate defeated Mr. Coburn’s amendment on a 54-45 vote, with nearly every Republican voting to reopen the White House and with almost all Democrats voting to back Mr. Obama’s decision.

“We’re focused on all the wrong things because it’s all about the next election,” Mr. Coburn said.

Now that it’s cleared the 2013 spending bill, the Senate turns its attention to the 2014 budget.

House lawmakers have already begun that debate, with Republicans defeating three Democratic alternatives on Wednesday. The main Democratic substitute was killed on a 253-165 vote, with even 28 Democrats balking at the tax and spending increases.

Hours earlier, Democrats tried to embarrass Republicans by refusing to cast votes on conservative Republicans’ alternative.

Rather than vote for or against the conservative budget, Democrats voted “present” — taking themselves out of the equation and making Republicans have to choose between backing the conservative blueprint, which calls for severe spending cuts, or to vote against it and risk the anger of tea party voters.

The final vote was 132-104 against the conservative plan, with 14 Democrats and 118 Republicans voting against it, and just 104 Republicans voting for it. If the conservative budget had won the vote, it immediately would have become the official House budget, replacing the plan drawn up by Rep. Paul Ryan.

Democrats wanted to force that situation, figuring the conservative budget would be too extreme for many moderate Republicans and therefore would ruin Republican efforts to pass a budget this year.