For nearly 15 years, a Texas death row prisoner has sought DNA testing that he claims will help to show he did not fatally stab an 85-year-old woman during a 1998 robbery.

On Thursday, the Supreme Court cleared the way for him to continue his legal challenge seeking DNA testing of crime scene evidence.

By a vote of 6 to 3, the court reversed a federal appeals court that had found Ruben Gutierrez was barred from bringing a lawsuit seeking the testing because he had failed to show that a state prosecutor would allow access to the evidence.

Justice Sonia Sotomayor wrote the opinion for the court, joined by the two other liberal justices, along with Chief Justice John G. Roberts Jr. and Justice Brett M. Kavanaugh. Justice Amy Coney Barrett joined in the court’s judgment and partially in the majority opinion, but filed a separate concurrence.

Justice Samuel A. Alito Jr. dissented, joined by Justices Neil M. Gorsuch and Clarence Thomas, who also filed his own dissent.

Last July, the justices made the extraordinary decision to spare Mr. Gutierrez’s life, halting his execution just 20 minutes before he was scheduled to be killed.