Simon Cowell & Baby Mama No More Hiding

8/24/2013 12:46 AM PDT

Simon Cowell & Lauren Silverman -- No More Hiding

Simon Cowell
, Lauren Silverman and their unborn spawn are making it official ... stepping out together for the first time IN PUBLIC since news of their relationship broke.

The pic was snapped in the South of France Saturday -- where the two are vacationing together amidst other rich people ... looking carefree and happy.

Probably has something to do with that whole nasty-divorce-settled thing.