Curtis Osborne had been set to die by lethal injection on Tuesday.
The court said it was granting an indefinite stay to Osborne because the U.S. Supreme Court is considering whether lethal injection constitutes cruel and unusual punishment.
The state Supreme Court issued a stay last week to convicted killer Jack Alderman for the same reason.
The stay for Osborne renders moot a clemency hearing he had scheduled for Tuesday before the state Board of Pardons and Paroles.
Osborne was convicted of killing Arthur Jones and Linda Lisa Seaborne, who were found in an automobile by the side of a dirt road. Both had been shot.
After investigation, Osborne was arrested, and eventually admitted opening fire in the car, claiming that Jones had reached toward the floor for a weapon, court records say.
However, the crime scene evidence suggested otherwise, court records say, adding that ballistics and fingerprint evidence was used to convict Osborne.
In Alderman's case, he was convicted of killing his wife in 1974 in Chatham County.
No one has been executed in Georgia since June, when a man convicted of killing his wife and two stepdaughters 20 years ago was given a lethal injection.
Several states have recently halted or delayed executions because of questions over the constitutionality of lethal injection procedures.