The bassist - who was previously engaged to supermodel Naomi Campbell and later split from Susie Smith, his girlfriend of 10 years, months before they were due to marry - had a son with an unnamed French woman.
His spokesperson confirmed the reports, but declined to give any details.
Friends close to the 50-year-old star told Ireland's Sunday Independent newspaper the rocker is delighted with fatherhood and having a child suits him considerably, despite the fact he once claimed he loved being in a band because it allowed him to be irresponsible.
He previously said: "I think one of the great things about bands is that they allow you to be irresponsible for longer - whether or not in the end that's a really healthy position to take.
"I guess I've been lucky in that I fucked about until my mid-30s and now I can have more of a balanced outlook. I think not having a family and kids, I know what I need."
Before the birth of his son Adam was the only member of the group - which also includes Bono, Larry Mullin Jr. and The Edge - not to have any children and Bono once admitted they "lived through him vicariously for a few years".
He added: I was hoping that he'd do something like buy a yacht and we could all hang out on it. Because all of us were too embarrassed."
Adam - who quit drinking alcohol in 1996 - has previously been linked to supermodel Helena Christensen.