The justices agreed to hear Henry Skinner's appeal. On March 24, they granted him a stay about an hour before his scheduled execution to give them more time to decide whether to take up his case.
In an order issued Monday, the Supreme Court said it decided to rule on the issue presented by his case. Arguments are expected to be heard in the upcoming term that begins in October.
Skinner's lawyers maintain that his rights under the civil rights law were violated by authorities' refusal to grant DNA testing after his conviction.
In the United States, post-conviction DNA testing has exonerated more than 250 people, including 17 prisoners who served time on death row, according to a group called the Innocence Project.
Skinner was convicted and sentenced to death for the murders of his girlfriend and her two adult sons on New Year's Eve in 1993 in the small town of Pampa, Texas. He has always maintained his innocence.
Skinner's attorneys are seeking DNA testing of key evidence from the crime scene, including a bloody towel, two knives and a man's windbreaker, and swabs from a rape kit.