The bill has broad bipartisan support, with 75 co-sponsors in the Senate. But its progress stalled Wednesday when Lee and Paul objected to a motion by Sen. Kirsten Gillibrand (D-N.Y.) to approve the measure by unanimous consent.

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Paul said Wednesday that he was objecting because any program that would last decades “should be offset by cutting spending that’s less valuable.” A spokesman for Lee said the senator was “seeking a vote to ensure the fund has the proper oversight in place to prevent fraud and abuse.”

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At a Capitol news conference, Gillibrand and her fellow New York Democrat, Senate Minority Leader Charles E. Schumer, hailed the agreement and predicted that the measure would pass overwhelmingly when it gets to the floor next week.

“We’ve already done a whip count. I am confident that the two amendments will be defeated and that we will get an overwhelming majority of senators for the vote to pass this bill and that the president will sign it,” Schumer said.

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The first responders, he added, “will soon finally have the peace of mind they deserve.”

Gillibrand voiced gratitude to the first responders and Stewart, describing the comedian as “a heartfelt advocate” for the legislation.

“His advocacy made such an extraordinary difference,” she said.

The House passed the Sept. 11 victim compensation bill 402 to 12 following the death of Luis Alvarez, a former detective with the New York Police Department who testified last month about the urgent need to replenish the fund. Officials say that money is fast running out, leading to payout reductions of as much as 70 percent for recent applicants.

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The fund provides money to those who have contracted diseases that have been linked to exposure to toxic debris as they worked at the site of the Sept. 11, 2001, terrorist attacks in Lower Manhattan. Lawmakers created the fund in 2011. It has paid about $5 billion to about 21,000 claimants. About 700 were for deaths that happened long after the attacks.

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Last month, McConnell was publicly rebuked by Stewart, the former host of “The Daily Show,” who lambasted lawmakers for dragging their feet on the measure.

A searing congressional hearing, featuring testimony from Stewart and the critically ill Alvarez, refocused public attention on the plight of the sick workers and the faltering fund.

On Wednesday night, Stewart turned his attention to Lee and Paul, accusing the two senators of “fiscal responsibility virtue signaling.”

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“It’s absolutely outrageous,” he said during an appearance on Fox News Channel.

Paul responded on the same network Thursday evening, calling Stewart a “guttersnipe” and blasting the legislation as a “completely irresponsible” measure that has received support only because senators are “overwhelmed” by the comedian’s celebrity.

“If Jon Stewart could read, maybe he’d read the bill and say, ‘Oh my God, who would vote for a bill that doesn’t have a dollar amount in it?’ ” Paul said.