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The decision marked a stark reversal for Mr. McConnell, who has collected nearly $460 million in pork-barrel projects over the past three fiscal year, according to Taxpayers for Common Sense, a nonpartisan spending-watchdog group.

Swarmed by reporters on Tuesday, Mr. DeMint said that Mr. McConnell proved he was a “good leader for the Republican Party” and credited him with getting most Republicans to buy into the idea.

“We have to stop the whole process, and hopefully, with the president at least saying he supports it, we can get the Democrats to come along, too,” he said.