The couple are thought to have welcomed a baby girl - Arlo Day Brody - on 4 August in Whittier, California.
The notoriously private are yet to confirm the birth, and according to People, reps for both stars had no comment to make.
Leighton was last publicly seen in July cradling her baby bump while out in Los Angeles. Reports that Leighton was pregnant surfaced in May after photos appeared online showing the actress and singer sporting a prominent baby bump.
Gossip Girl star Leighton met The O.C. star Adam on the set of The Oranges in 2011. They tied the knot in early 2014 but have never really commented on their wedding.
However, in an interview with Nylon in November, Leighton hinted that starting a family was on the cards.
"Most people I know are starting to get married. It'll be kids next," she said. "I like the grown-up stuff. I like having a house. I've got dogs."
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