Brad Pitt was at Angelina Jolie's side as she gave birth to a son and daughter Saturday at the Fondation Lenval hospital in Nice, France, PEOPLE has confirmed.

"The babies are doing well. The operation went just perfectly," Dr. Michael Sussmann told PEOPLE Sunday. "Angelina is in very good spirits. Brad Pitt was at her side. He was there and all was well."

Sussmann confirmed the Nice Matin newspaper report that Jolie gave birth to a boy, Knox Leon, and a girl, Vivienne Marcheline, by Cesarean section on Saturday night. Knox weighed 5.03 lbs, and Vivienne 5 lbs.

The twins are the fifth and sixth children for Jolie, 33, and partner Brad Pitt, 44, who are already parents to Maddox, 6; Pax, 4; Zahara, 3; and Shiloh, 2.

"The mother and father are very, very happy," Sussmann told PEOPLE.

The doctor told the Associated Press that the Cesarean was moved forward "for medical reasons" so that the babies could be born "in the best conditions." He also said Jolie is expected to stay in the hospital for a few days.

On July 1, Jolie checked in to the Fondation Lenval hospital to await the births. The family has been living in the south of France since this spring.

Jolie recently told Entertainment Weekly that she was surprised - but not daunted - when she learned she was having twins. "It did shock us, and we jumped to six [children] quickly," she said. "But we like a challenge."

Jen's middle name is Lynn?