A restrictive law on voting in Kansas championed by Kris W. Kobach, the secretary of state, was struck down on Monday by a federal judge who said Mr. Kobach had failed during a trial to show evidence of widespread voter fraud.

The ruling was a blow to Mr. Kobach, a Republican who has emerged as a national figure on voting limits, a candidate for governor of Kansas and an ally of President Trump in part by claiming that large numbers of noncitizens have cast ballots in American elections. Experts on election law say that there is no evidence that voter fraud is a pervasive problem.

For Kansas voters, the decision means that in elections this fall, people will not be required to provide proof of their citizenship in order to register to vote, as required under a Kansas law passed in 2011.

Federal District Judge Julie A. Robinson of Kansas, who presided over the trial earlier this year in which Mr. Kobach represented himself, said in her 188-page ruling that while there was evidence of a “small number of noncitizen registrations in Kansas, it is largely explained by administrative error, confusion, or mistake.”