Simon Cowell -- OFFICIALLY A DAD ... GF Lauren Silverman Gives Birth

4:25 PM PT -- the baby's name is ERIC ... after Simon's father.

Simon Cowell is officially a father ... his pregnant GF Lauren Silverman has finally popped in NYC.

36-year-old Silverman just gave birth to a healthy baby boy at Lenox Hill Hospital -- after Simon flew across the ocean from London to be by her side for the delivery. As we reported, Lauren went into labor early this morning and Simon hopped on a plane as soon as he heard.

We're told the baby was born at 5:45 PM EST at 6 pounds, 7 ounces.

The boy is 54-year-old Simon's first child and Lauren's second -- she also has an 8-year-old son with her ex Andrew Silverman.

So far, no word on the baby's name.