The American actress and French actor, both 47, have named their son Maceo-Robert, a source close to the couple confirms to PEOPLE.
The "Maceo" part is said to be of Spanish origin – Martinez's father, Robert Martinez, was a Spanish boxer – and means "gift of God."
TMZ.com was the first to report the name.
The hyphenated name gives the boy one foot in the old world and one in the new – perfect for the transatlantic couple, who "decided on a name that's great in both the U.S. and France," a source close to Martinez recently told PEOPLE.
Berry, who won the Oscar in 2002 for Monster's Ball, also has a 5½-year-old daughter, Nahla Ariela, from her previous relationship with Gabriel Aubry.