"Angie and baby are fantastic," a source close to the couple told PEOPLE on Sunday. "Brad was at her side during the birth."
The child was born Saturday, May 27, at night, in the African country of Namibia.
Shiloh – which in one longstanding translation from the Bible has come to mean "the peaceful one" – was born in The Cottage Hospital in Swakopmund.
Jolie is expected to remain in the hospital for a couple days before returning to the couple's sea-side Namibian retreat.
Meanwhile, Namibia's Environment and Tourism Deputy Minister, Leon Jooste, explained: "Shiloh Nouvel Jolie-Pitt will, according to Namibian law, be allowed to obtain Namibian citizenship, if the parents should choose to do so." He added that discussion of the child's passport will be taken up with her parents "at a later stage."
The new parents, along with kids Maddox, 4, and Zahara, 1, have been staying at the luxury resort in Namibia since April, where their personal security team and Namibian police have shielded them from paparazzi.
Last week, the Namibian government said foreign journalists wishing to cover the birth must have written permission from Pitt and Jolie to enter the country and obtain a work permit.
The new girl is the couple's first biological child. Jolie adopted Maddox from Cambodia in 2002 and Zahara from Ethiopia last year. Pitt is in the process of legally adopting both children.
The baby is the latest step in a life-changing year during which Pitt and Jolie went from costars (in last summer's slick crime thriller Mr. and Mrs. Smith) to supercouple to, finally, Mommy and Daddy.
Word that Angelina was pregnant first broke on Jan. 11, when representatives of both stars confirmed the baby news to PEOPLE after the actress and U.N. goodwill ambassador had confided to a charity aid worker in Santo Domingo, Dominican Republic: "Yes, I'm pregnant."
The pregnancy news – which came shortly after Pitt's announcement that he planned to adopt Maddox and Zahara – might have been a media bombshell, but it was a natural progression for the pair who put nest-building on the fast track.
Jolie, 30, had long spoken about expanding her family, and told PEOPLE last October that she would like to adopt again. "There's something about making a choice, waking up and traveling somewhere and finding your family," she said.
"I think he'll be wonderful (with the new baby)," Pitt's Assassination of Jesse James costar, Mary-Louise Parker, told PEOPLE earlier this year. "He's a great guy, conscientious and kind. I'm happy for both of them."