In a unanimous decision, the five-member court voted to let stand lower court decisions denying Tyson's request.
The defendant stood with his head bowed but showed no reaction as the presiding judge, Heiner Muckenberger, read the findings of the five-member court.
Overturning a lower-court ruling that had provoked wide controversy, the five-member court, in a one-line decision, allowed the girl to travel.
In a surprise development, a court official said a rehearing had been set for a five-member court on Sept 9.
The five-member court was divided, 2-2.
The five-member court, which includes one woman, has 60 days to decide.
The five-member court in Bloemfontein also overturned Mrs. Mandela's conviction and one-year sentence on a related charge of being an accessory to assault.
The unanimous ruling by the five-member court overturned a decision by Acting Justice Walter M. Schackman of State Supreme Court in Manhattan.
On 31 December 1975, the five-member court in Athens convicted 20 of the 32 accused and held Ioannidis as the only person morally responsible for the events.
The five-member court delivered the ruling to the two governments this morning but officials agreed to delay public disclosure of the decision until Monday.