"The mother and her babies are doing very well," the statement said.
AP Television cameras outside the hospital filmed a large white van with tinted windows leaving one of the hospital's back entrances at that time. No other vehicles left the hospital for the next several hours.
Jolie checked into the hospital on July 2 and delivered the babies by Cesarian section 10 days later. Knox Leon weighed 5.03 pounds and the Vivienne Marcheline 5 pounds.
The Nice-Matin newspaper has reported that an unnamed U.S. publication has paid $11 million for exclusive rights to the first photos of the Jolie-Pitt twins and that the proceeds will go to charity.
In its statement, the Fondation Lenval thanked the Jolie-Pitt couple for choosing the hospital.
It also thanked the hospital staff "for the attention and care they paid to the family" and the "journalists around the hospital who did their best to respect their privacy."
Jolie and Pitt have four other children - Maddox, 6; Pax, 4; Zahara, 3; and Shiloh, 2. Before Jolie checked into the hospital in Nice, the family set up a household in Correns, 60 miles away.
Jolie's latest film, the thriller "Wanted" with James McAvoy, has wracked up $112 million in sales in its first three weeks in the United States.
