It is understood Sir Mick flew 3,500 miles from London to New York in a last-minute dash to be by Hamrick's side for the birth.
The source added: 'He made sure he was there. He wouldn’t have missed it for the world.'
A statement released on behalf of the couple said: 'Melanie Hamrick and Mick Jagger's son was born today in New York and they are both delighted.
'Mick was at the hospital for the arrival. Mother and baby are doing well and we request that the media respect their privacy at this time.'
The news comes after it was revealed that the Rolling Stones are on course for this week's UK Number One album.
The rockers’ latest LP ‘Blue & Lonesome was way ahead in the midweek charts.
Sir Mick is understood to have given Melanie a $150,000-a-year maintenance deal to help the ballerina raise their child.
He is also believed to be helping the New Yorker find a home in the US to raise the boy.
The $150k will be paid until the child is 18, with school fees also paid for. After secondary school a trust will help cover further education and expenses, which could be as much as £500k.
Even though Mick and Melanie are not planning on getting married, they plan to co-parent given his hectic work schedule.