WASHINGTON—The Supreme Court on Friday allowed Texas to continue to enforce a law requiring voters to show identification at the polls, a setback for civil rights challengers who said the law could make it difficult for a sizable number of minorities to cast a ballot in November.

The court, in a brief written order, declined to disturb a lower-court action that has allowed Texas to use its voter-ID law while litigation over the measure continues, including during its recent primary election in March.

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