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Mr. Kennedy, Mr. Kyl and others involved in the grand bargain have been successful in fending off most challenges to the bill, and leaders on both sides have said any deal-breaker amendments that pass can always be dumped in a House-Senate conference.

But that is exactly what worries conservatives such as Mr. DeMint, and prompted his pledge to block a conference.

The Bush administration is making a full-court press for the bill, rallying grass-roots and business group leaders at a meeting in the White House complex yesterday, while Homeland Security Secretary Michael Chertoff visited a construction company in the District to watch a demonstration of what will become the employment-verification system.

He said the pilot program is working well, but has a flaw — it can’t weed out illegal aliens who are using real Social Security numbers stolen from someone else. He said the Senate bill changes that.

“Right now the law forbids us from getting into Social Security to determine whether we can verify in fact the genuine owner of the number and whether we’ve got two people using the same number at the same time,” he said. “The only way to get that is for Congress to pass this bill.”

In his remarks at the prayer breakfast yesterday, Mr. Bush made a pitch for passing a bill that addresses many parts of the immigration issue: better enforcement, a temporary worker program, legalization for illegal aliens and assimilation for new immigrants.

“That’s what has always made our nation strong. People in America must have confidence in this country to help people assimilate,” he said.