First an Emmy, and now a baby.

"Will & Grace" star Debra Messing, 35, is a new mother, having given birth to a "healthy, beautiful" boy Wednesday night in Los Angeles, her rep confirms to PEOPLE. The baby, born at Cedars-Sinai Medical Center, weighed 5 lbs. 14 oz., and was 19 in. long. No further details, such as the child's name, have been released.

The dad is Messing's husband of four years, actor-screenwriter Daniel Zelman, 36, and it's the first child for both.

"The baby's healthy, Debra's healthy," the rep tells PEOPLE. "It's an extraordinary time for them. Everything is great. They're thrilled."

As the pregnant Messing moved closer to her due date, shooting her sitcom was becoming more challenging. "We are running out of ways to hide Debra's stomach," said her costar Eric McCormack, 40. "At first the refrigerator door was good enough, but now we have to have trucks drive through scenes."

Ultimately, Messing was ordered to stay off her feet. She was absent from the set for the better part of March and won't appear in the series' final episodes of the season, including the April 29 finale guest-starring Jennifer Lopez. The episode reportedly centers on the marriage of Karen (Megan Mullally).

Speaking to PEOPLE early this year about being pregnant. Messing, who first met Zelman when they were both students at New York University (and then dated for eight years), said, "I want my child to be like my husband. He is sensitive, really intelligent, really funny ... but solid. He has a good sense of perspective."