Sources tell TMZ ... the Oscar-winning actress gave birth Saturday, March 13 in the L.A. area. No word yet on if they had a boy or a girl, but we're guessing Emma and hubby comedian/writer/director Dave McCary are thrilled either way.

We last saw Emma out about last month in L.A. ... looking elegant, but very ready for the baby to arrive.

Emma and Dave met when she was hosting 'SNL' in 2016, and he was directing a taped sketch for the episode. They kept their relationship under wraps for quite a while ... until they went public by attending the 2019 Screen Actors Guild Awards together.

Dave eventually popped the question in December 2019, which they announced on social media -- Emma looked pretty damn proud, showing off the huge rock, and Dave captioned his post with a simple double heart emoji.