Sarah Jessica Parker gave birth to a baby boy early Monday morning in Manhattan's Lenox Hill Hospital, the New York Post reported Tuesday.

The child weighed in at 6 lbs. 8 oz. and measured 19.5 in., says Parker's publicist, Ina Treciokas, who tells PEOPLE.com, "Mother and baby are doing very well. Everyone is healthy."

But Treciokas disputes the Post's report that the baby is named David Perkins Broderick. (The paper said the name was on a placard posted outside the actress's hospital room.)

"It's inaccurate to say the baby's name is David," Treciokas says. When asked to supply the baby's name, Treciokas replied, "I don't have one, and I don't know if there is one yet."

The Post quoted unnamed sources saying the baby was born around 3 a.m. and that the "Sex and the City" star's delivery was relatively smooth, after "pushing for just an hour."

This is the first child for Parker, 37, and her actor husband Matthew Broderick, 40, who have been married since 1997. Broderick became a daddy-to-be shortly after wrapping his run in Broadway's "The Producers." He also stars in an upcoming made-for-ABC-TV version of the musical classic "The Music Man."

The Post also said that Cynthia Nixon, one of Parker's costars on HBO's "Sex and the City" -- and who is also expecting a baby -- was one of the few visitors to pay her respects to Parker on Monday and that her visit was brief.

HBO (which, like PEOPLE, is part of AOL Time Warner) has issued earlier statements saying Parker, who is also executive producer of the Emmy-winning series, will begin filming the show's new season next February. The current season had a truncated run due to Parker's pregnancy, which did not figure into the storyline.

The paper also reports that Broderick and Parker were last seen on the town together Saturday night, when they had a quick, late-night snack at a downtown Asian restaurant after attending a performance of the new musical "A Man of No Importance" at Lincoln Center.

Only last Thursday, Parker made a public appearance to honor her favorite shoe designer, Manolo Blahnik, at the 19th Annual Night of the Stars for The Fashion Group International dinner.

Now it's time for midnight feedings.