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DES MOINES — For the first time, Senator Ted Cruz of Texas took the top spot in an early-state poll, pulling ahead of Donald J. Trump in Iowa in a survey released on Monday by Monmouth University.

Mr. Cruz, the beneficiary of a crucial endorsement by a hard-right Iowa congressman and the precipitous decline of Ben Carson, was supported by 24 percent of likely Republican caucusgoers in Iowa, where the first nominating contest is 56 days away. Mr. Trump had 19 percent, Marco Rubio had 17 percent and Mr. Carson was at 13 percent.

The poll shows a big shake-up since the last Monmouth poll in October, when Mr. Carson led. He has since fallen by 19 percentage points and Mr. Cruz has jumped up by 14 percentage points.

Mr. Trump and Jeb Bush, who was the favorite of 6 percent of likely Republican caucusgoers, were largely unchanged. Mr. Rubio’s support increased seven percentage points since October.

Mr. Cruz, who has ardently sought the support of evangelicals, the largest bloc of Republican caucusgoers, was backed by 30 percent. Evangelicals, whose support earlier put Mr. Carson put him atop the previous Monmouth poll, are deserting the retired surgeon: only 15 percent support Mr. Carson, whose lack of foreign policy expertise has cost him as the nominating race has turned to fears of terrorism. Mr. Carson’s favorability rating, once off the charts, fell to 67 percent positive from 84 percent.

Nearly one in five Republicans polled said Mr. Cruz’s endorsement by Representative Steve King of Iowa, a fierce opponent of illegal immigration, makes them more likely to support Mr. Cruz.

The shake-up in the field shown in the poll is unlikely to be the last before the Feb. 1 caucuses. In past races, about 40 percent of caucusgoers have made up their minds in the days before voting. The Monmouth survey showed just one in five voters are set on their candidate, and another 49 percent have a strong preference.

Marco Rubio, Switching Focus, Aims to Halt Ted Cruz’s Momentum In interviews, speeches and in stealthier ways, Mr. Rubio has abruptly changed the course of his campaign to zero in on Mr. Cruz and reverse his gains among conservative voters.