Lauren went into labour two weeks early and was taken to Lennox Hill hospital in New York on Friday morning.
Simon arrived at the hospital – after abandoning filming for Britain’s Got Talent – around 2pm, and told waiting photographers that Lauren was doing well.
The proud dad tweeted confirmation of the baby's arrival in the early hours of this morning. He wrote: "Very happy to say Eric was born at 17.45pm. Healthy and handsome.
"Lauren in great shape. Thanks for all of your kind messages. Named after my dad."
Ex girlfriend and close pal Sinitta tweeted her congratulations, writing: "All is well and baby boy says 'parents are well and CUTE!' Happy Valentines day BABY!! #JustInTimeValentine!"
Earlier in the evening, Sinitta hinted that she had already been picked to be Godparent to Simon and Lauren's child, posting: "Sitting in. Fancu bar in london with a very handsome man waiting to hear my God son has been born, bit tipsy,but not ruined..too late to fly."
Yesterday, other friends of the couple tweeted their excitement about the arrival of the X Factor moguls first child, with Simon’s BGT co-judge David Walliams writing: “I am wishing @SimonCowell and the lovely Lauren Silverman all the best for today for the arrival of Baby David.”
Congratulations, Simon and Lauren!