By TRIP GABRIEL

The poll suggests that the Texas senator’s campaign is gathering momentum and that there could be a long Republican nomination fight ahead.

Donald J. Trump, who continues to lead most national polls, was second in the Iowa poll, with 21 percent naming him as their first choice, compared with Mr. Cruz’s 31 percent.

Ben Carson, after leading an October survey by the same pollster, commissioned by The Des Moines Register and Bloomberg Politics, tumbled to 13 percent. Senator Marco Rubio of Florida had 10 percent. No other candidate had double-digit support.

The survey, by one of Iowa’s most respected pollsters, comes as Mr. Trump is increasingly hoping that a win in Iowa, whose caucuses are less than eight weeks away, will be a lightning strike on his way to the nomination.

“If we win Iowa, I think we run the table,” Mr. Trump said on Friday at a rally in Des Moines. He is far ahead in polls of the other early-voting states.