WASHINGTON, June 6 — The proposed immigration-overhaul bill survived a crucial test this afternoon as the Senate rejected a measure that would have made it much harder for many illegal immigrants to eventually achieve legal status. By a vote of 51 to 46, the senators defeated a proposal to bar legal status for aliens who disobey deportation orders or who engage in identity fraud. The proposal was made in the form of an amendment, offered by Senator John Cornyn, Republican of Texas, to an overall immigration bill that is being debated.
The Cornyn amendment, one of several voted on today, was considered especially important, because many of the estimated 12 million illegal immigrants in the United States commit some kind of identity fraud, for example when they use bogus Social Security cards in trying to get jobs.
Senator Edward M. Kennedy, Democrat of Massachusetts and a sponsor of the overall bill, argued against the Cornyn amendment, declaring that to approve it would be “undermining the basic core” of the legislation. Ten of Mr. Cornyn’s fellow Republicans voted against his amendment, while nine Democrats voted for it as the immigration issue continued to cut across party lines.
Mr. Kennedy offered an amendment of his own, which was approved by 66 to 32, to deny legal resident status to gang members, sex offenders and people who are guilty of domestic violence — but not to those who engage in document fraud. In effect, Mr. Kennedy’s amendment offered political cover to lawmakers who favor the overall legislation but do not want to appear too easy on immigrant law-breakers.
After the votes, Senator Harry Reid of Nevada, the Democratic majority leader, said he thought the Senate was “making good progress” on the legislation, and that he hoped to gain cooperation in whittling down the number of amendments remaining to be voted upon.
Later, the Senate rejected another amendment that the bill’s backers considered too harsh on immigrants. That amendment, offered by Senator Jim DeMint, Republican of South Carolina, would have required undocumented aliens to get health insurance before they could be legalized. It failed by 55 to 43.
The Senate voted to keep a requirement that most “guest workers” go home for a year between each of three two-year stays in the United States. An amendment that would have removed that requirement was defeated by 57 to 41. The amendment was offered by Senator Jeff Bingaman, Democrat of New Mexico.
Mr. Reid’s assessment that the Senate was making progress was important, because he said on Tuesday that the chamber would vote Thursday on whether to limit debate on the bill, a process called cloture that requires 60 votes to succeed. If the cloture vote fails, the bill could be blocked indefinitely by a filibuster. Mr. Reid said he would pull the bill from consideration if he fails to get the necessary votes.
The majority leader said he wanted to complete work on the legislation this week, and he suggested that Republicans were trying to stall the bill with amendments.
“When is enough enough?” he asked, asserting that Republicans were looking for excuses to kill the bill. His announcement provoked an outcry both from Republican supporters and Republican opponents of the compromise bill, who said the Senate needed more time.
Senator Jon Kyl of Arizona, the chief Republican architect of the bill, said “it would be a big mistake” to try to invoke cloture this week.
“A motion to cut off debate would be an extreme act of bad faith,” Mr. Kyl said, and he asserted on Tuesday afternoon that “we are not anywhere near finishing this bill.”
The Senate Republican leader, Mitch McConnell of Kentucky, said, “The overwhelming majority of our conference would insist on having additional days to make sure that all of our important amendments have been given an opportunity to be considered.”
Even Senator Mel Martinez, Republican of Florida, a strong supporter of the bill, said, “I would not support cloture at this point because I don’t think that enough of our members have had an opportunity to have their amendments heard.”
Behind the scenes, senators were trying to work out agreements on what amendments to consider, so the legislation would not fail on a motion to limit debate.
Democrats have offered amendments to promote unification of families, by providing more green cards for relatives of American citizens and lawful permanent residents who want to come to the United States. For their part, Republicans have offered amendments to deny legal status to illegal immigrants who have defied deportation orders.
Mr. Reid said that if he could not muster the votes needed for cloture, he would move on to other matters — a vote of no confidence in the attorney general, Alberto R. Gonzales, and then energy legislation.
Mr. Reid said he saw only a tiny possibility that the Senate might return to immigration at a later date, but he added, “I never say never.”
David Stout contributed reporting for this article.