Julia Roberts's publicist says "mother and babies are doing great," after the Erin Brockovich Oscar winner gave birth to twins – a boy and a girl – in a Los Angeles hospital on Sunday, as PEOPLE first reported.
Their names are Phinnaeus Walter Moder and Hazel Patricia Moder. Weights and other details about the new arrivals have not been announced.
Roberts, 37, and Moder, 35, welcomed Phinnaeus and Hazel at about 3 a.m. Sunday morning, a source tells PEOPLE. They are the couple's first children, whose arrival date was not expected until early January.
Late last month, Roberts was admitted to an L.A. hospital after experiencing a series of early contractions. The condition was not serious, and the contractions eventually stopped, but doctors advised bed rest until she gave birth.
Roberts and Moder have kept a low profile since marrying on July 4, 2002, at her 82-acre ranch in Taos, N.Mex. The decidedly private couple split time between their homes in Taos, Venice, Calif., and New York City.
Twins, it turns out, run in Roberts' family: Her great grandmother and a pair of cousins are twins.
The Pretty Woman star has two movies opening next month: Ocean's Twelve, with George Clooney and Brad Pitt, and Closer, an adult drama directed by Mike Nichols.