“Endorsements alone don’t guarantee victory, but Palin’s embrace of Trump may turn the fight over the evangelical vote into a war for the soul of the party,” he said.

Mrs. Palin could amplify the news media-circus aspects of Mr. Trump’s candidacy: She too is a reality television star accustomed to playing to the cameras and often accused of emphasizing flash over substance.

And while Mr. Trump has already shown the ability to garner wall-to-wall cable-news coverage, Mrs. Palin’s involvement in his campaign could help him deprive Mr. Cruz of attention in the homestretch to the caucuses.

As rumors circulated that the endorsement was about to happen, Mr. Cruz offered praise for his former political ally after an aide to the senator mocked the pending endorsement earlier Tuesday. “I love Sarah Palin,” the senator told reporters in New Hampshire. “Sarah Palin is fantastic. Without her friendship and support, I wouldn’t be in the Senate today. So regardless of what Sarah decides to do in 2016, I will always remain a big, big fan of Sarah Palin.”

As word of Mrs. Palin’s endorsement trickled through the Hansen Agriculture Student Learning Center at Iowa State University, the reaction from supporters of Mr. Trump who braved snow and frigid temperatures to see the candidate was mixed. Backers of Mr. Trump filled a warehouse-style building with a dirt floor that is sometimes used for tractor shows, but most said that it was the candidate that they cared about, not his new endorsement.