The boy's name is Pax Thien Jolie, born in November 2003, according to the orphanage where he has lived. His name means Peaceful Sky – Pax is peace in Latin and Thien is sky in Vietnamese.
"Thank you for taking such wonderful care of my son," the Oscar-winning actress wrote in the visitors' book of the Tam Binh orphanage, according to Reuters. "I am sure he will miss your beautiful, caring staff as much as you will miss him."
Jolie's rep says she took custody of the child on Thursday in Vietnam. They will remain in the country for about a week while the child's American passport is processed, the rep says.
The boy had been living at the Tam Binh orphanage outside Ho Chi Minh City ever since he was abandoned at a hospital as a baby, according to adoption officials. Shortly after he arrived at the facility, staff attempted to locate his birth parents, but to no avail.
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"Brad very much wanted to be there but is under contract working on a film in Los Angeles and as a result couldn't go," Trevor Neilson, Jolie's adviser on international affairs, tells PEOPLE.
About two dozen children in traditional ao dai formal garb greeted them and offered flowers at the orphanage. Jolie left carrying the boy to a waiting van.
"He is in good health," orphanage director Nguyen Van Trung told the Associated Press, adding that the boy gets along with the other children and likes playing soccer. "He is a little bit shy."
Trung also said Jolie's adoption was swiftly processed because the boy is older and his files were already nearly complete.
Because Vietnamese law makes it difficult for unmarried couples to adopt, Jolie applied solo – but she and Pitt together will welcome their new Vietnamese child to their family, which along with Maddox includes Zahara, 2, and 10-month-old Shiloh. The adoption was a "process that began shortly after Shiloh was born [in May]," a source told PEOPLE in a recent cover story.
After adopting Maddox from Cambodia and Zahara from Ethiopia, she mused to Good Morning America in December that it might be a good idea to "balance the races so there's another African person in the house for Z [and] another Asian person in the house for Mad."