We've obtained a copy of the birth certificate -- filed with the L.A. County Dept. of Public Health -- and it says Montgomery Moses Brian Baron Cohen was born on March 17.
It just came out last week they'd had the baby -- but they didn't say the name, sex or birth date.
MMBBC is the third child for Isla and the 'Borat' star. They have 2 daughters ... Olive, 7 and Elula, 4.
As for all the names, Isla's dad's name is Brian -- but we're at a loss on Montgomery and Moses ... outside of the 'Ten Commandments' guy.