E! News confirmed on June 6 that the human rights attorney had given birth to their first children, twin son and daughter, Alexander and Ella.
The family of four spent "a little over three days" at the hospital before heading to their home in England, where they have been joined by their nearest and dearest, such as Amal's sister Tala Alamuddin.
"It was a longer birth than expected but the babies are healthy and beautiful," the source said. "They were able to keep a low profile at the hospital and even have family and friends visit."
"They're elated. I don't think you've ever seen a happier pair," the source added. "There's a positive energy in the house and everyone around them, including all staff, who are in great spirits too."
George's father Nick Clooney said on Good Morning Britain last week that the twins are "beautiful." He said "George will be an absolutely wonderful father" and that "Amal will be—and already is—a magnificent mother."