Sources close to the situation tell TMZ ... Cassie gave birth Friday at an L.A. County hospital, and the couple named their first child Frankie -- Frankie Fine.

We're told their bundle of joy weighed in at 8 lbs., 4 oz and measured a smidge over 21 inches long.

As we've reported, Diddy's ex and Alex announced they were expecting in June, got engaged a couple months later ... then got hitched in Malibu in September.

If you recall ... there was some drama when the 2 began dating last year, because Alex was her former trainer. Any potential bad blood was washed away by the time they revealed their pregnancy news, though ... Diddy gave them a shout-out, wishing them nothing but love and happiness.

The well-wishes seem to be working. Congrats to Cassie, Alex and, of course, Frankie!!!