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Despite the drama Wednesday, the increase in the debt limit was much easier than previous votes — chiefly because Republicans surrendered early in hopes of attaching conditions.

The outcome was assured — the only question was which Republican would have to cast the politically perilous vote. During the hourlong process, Republicans gathered in small groups on the Senate floor to discuss the matter.

Senators said four Republicans were willing to join all 55 Democrats, but that left them one vote shy of the 60 needed to break the filibuster.

Ultimately, Mr. McConnell came to the well of the chamber and cast his vote as the crucial 60th supporter.

Sen. John Cornyn of Texas, the second-ranking Republican, then switched to join Mr. McConnell in opposing the filibuster. He and other allies then went to try to persuade other Republicans to join them in a show of solidarity.

“All right, let’s go — come on down,” Sen. Bob Corker, a Tennessee Republican who voted “yes” to break the filibuster, told his colleagues.

But few followed. The final tally was 67-31. Minutes after breaking the filibuster, the debt increase was approved on a party-line 55-43 vote.

Mr. Corker criticized Mr. Cruz for forcing such a politically dangerous vote.

“Cloture is to end debate. There was no alternative offered. We knew the House could only pass a clean [debt increase], so why do you debate it any longer? It’s either a yes-or-no vote,” he said.

House Republican leaders passed the debt increase on a 221-201 vote that relied almost entirely on Democrats. Just 28 Republicans voted for it.

The GOP was fractured on the issue, and many were spooked after being blamed for the partial federal government shutdown in October. Democrats were united against any concessions, convinced that Republicans would be blamed for a breach of the debt ceiling.

Republicans said they agreed to the increase to avoid detracting from their message against Obamacare, which they believe will help them regain control of the Senate in November’s elections.

“Our focus really continues to be on Obamacare and other issues,” said Sen. John McCain, an Arizona Republican who initially supported the filibuster but switched his vote after Mr. McConnell voted. Mr. McCain called Mr. McConnell’s vote “a very courageous act,” given his tough re-election race.

Mr. McConnell didn’t issue any statement about his vote.

Afterward, the Senate wrapped up business and went on recess for 11 days.