Sean "Diddy" Combs' let one of his reality show contestants slip through his hands ... because she's got real talent -- FILMING HERSELF GIVE BIRTH IN A CAR.
Brianna Smith -- who lost Season 1 of "I Want To Work For Diddy" -- was racing to a Houston hospital with her hubby in the middle of the night when the kid starting popping out.
She had her phone out and made the most of the moment ... shooting the horrendous beautiful moment of birth.
The kid's name is Theodore Roosevelt Smith IV ... which is perfect, because he's already had a rough ride.
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