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Two months before the Iowa caucuses, Donald J. Trump and Hillary Clinton are showing no signs of losing steam.

A new national poll from Quinnipiac University finds the leading candidates solidifying their positions in the races for the Republican and Democratic nominations, fending off challenges from rivals such as Ben Carson and Senator Bernie Sanders of Vermont.

On the Republican side, Mr. Trump has benefited from recent stumbles by Mr. Carson, the retired neurosurgeon whose rise has been stymied by questions about his biography and his knowledge of foreign policy.

A month ago, the two were deadlocked, but the survey results released Wednesday show Mr. Trump clearly in first place with 27 percent of Republican voters. Mr. Carson has dropped to third place with 16 percent, having been overtaken by Senator Marco Rubio of Florida at 17 percent. Senator Ted Cruz of Texas is tied with Mr. Carson at 16 percent, having risen from 13 percent in a previous poll.

Hoping to strengthen his foreign policy credentials and to educate himself about foreign affairs, Mr. Carson made an impromptu trip to Jordan over the weekend to see the Syrian refugee crisis firsthand. On Wednesday, he heads to South Carolina, and later this week he will go to Iowa, where he will seek to solidify his support among evangelical Christian voters.

Mr. Trump’s endurance comes as he continues to face questions about his honesty and as he stirs controversy with his ideas about aggressive surveillance of Muslims, his proposal to reinstitute waterboarding and his mockery of a New York Times reporter with a physical disability.

Many Republican leaders are actively fretting about Mr. Trump’s continued strength, but they remain wary of attacking him directly out of fear that they will have to endure his vicious counterattacks. This week, Mr. Trump called Gov. John R. Kasich of Ohio a “lunatic” and ridiculed Gov. Chris Christie’s record in New Jersey after the two rivals for the nomination were openly critical of him.

“It doesn’t seem to matter what he says or who he offends, whether the facts are contested or the ‘political correctness’ is challenged, Donald Trump seems to be wearing Kevlar,” Tim Malloy, assistant director of the Quinnipiac University poll, said in a statement on Tuesday. “The G.O.P., 11 months from the election, has to be thinking, ‘This could be the guy.’ ”

For Democrats, this appears to be good news. Quinnipiac’s poll shows both Mrs. Clinton and Mr. Sanders beating Mr. Trump handily in head-to-head matchups. At the moment, Mrs. Clinton appears to be the clear favorite, having widened her advantage against Mr. Sanders: The poll shows her ahead by a margin of 60 percent to 30 percent among Democratic voters.

Mr. Sanders lost some momentum after the Democratic debates, where Mrs. Clinton performed well. While voters still have doubts about her honesty, questions about Mrs. Clinton’s use of a private email server as secretary of state have subsided for the time being.

Mrs. Clinton’s momentum has improved her standing in face-offs against all the leading Republican contenders, the poll shows. However, Mr. Rubio, who has been rising steadily, fares the best against her, trailing by just one percentage point.

The Quinnipiac poll had a margin of error of four percentage points for Republicans and Democratic voters, and three percentage points over all.