The court's decision means Warren Lee Hill will not likely be scheduled for execution before the state's current supply of a lethal injection drug expires next week.
Hill was one of two Georgia inmates to be executed this week. The other was also granted a stay, although the state has appealed.
Hill was convicted in the 1990 beating death of fellow inmate Joseph Handspike. At the time, Hill was serving a life sentence for killing his girlfriend.
The state says the defense has failed to meet its burden of proving beyond a reasonable doubt that Hill is mentally disabled.