June 28, 2004 12:00 PM

Saturday, June 12, was all set to be a romantic fifth-anniversary celebration for Courteney Cox Arquette and David Arquette. Four days earlier, Arquette had purchased a five-carat diamond ring at Victoria Jewels in Beverly Hills for his wife, who was pregnant with their first child. But the pair soon got a priceless gift: Just after 2 p.m., Cox went into labor, a week ahead of schedule. Fifteen hours later, with Arquette by her side, Cox gave birth to a healthy 6-lb. girl at Cedars-Sinai Medical Center in L.A. “It’s the best present they could have hoped for,” says a source close to Arquette. “It’s been a long haul.”

The couple have been trying to start a family since 2001, but they suffered several miscarriages and failed attempts at in vitro fertilization. “I don’t say it’s a walk in the park,” Cox said last October. “But what are you going to do? We just try again.” Parents at last, the ecstatic pair named their daughter Coco. “It’s not a name that runs in either of their families,” says the source. “But Courteney’s family calls her Cece. Maybe that had something to do with it.”

The evening of June 13, the whole family went home to Hollywood Hills. “I’m gonna have a ton of fun being a dad,” Arquette, 32, said last month. “I can’t wait to read stories and play games.” Friends of Cox, who turned 40 two days later, expect her to adjust easily to motherhood. “Courteney is warm, caring and generous,” says actress Christina Applegate. “All the perfect qualities of a mother.”

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