Leighton Meester & Adam Brody -- We Had a Girl ... Named Arlo

Leighton Meester and Adam Brody have been parents a lot longer than people knew ... and TMZ has uncovered the name and gender of their down-low baby.

According to the birth certificate obtained by TMZ, Meester gave birth to a girl on August 4 ... not last week, as has been reported.

The girl's name is Arlo Day and she was born just after midnight at a hospital located about 30 miles away from Cedars-Sinai Medical Center, the usual celebrity hospital of choice.

The couple was married in February 2014 in a very hush-hush ceremony.