The baby boy was born last week to Adele Adkins and her boyfriend Simon Konecki, said the source, whose name was not reported.
"Adele and Simon are ecstatic at their new arrival," the source said. "She was over the moon that they were expecting their first child together."
Adele's rep had no comment on the birth Saturday.
"We are not releasing a statement at this time. No comment," he said.
Adele announced the pregnancy four months ago on her official website.
"I'm delighted to announce that Simon and I are expecting our first child together. I wanted you to hear the news direct from me, obviously we're over the moon and very excited but please respect our privacy at this precious time. Yours always, Adele xx," she wrote.
The six-time Grammy Award winner denied rumors in May she was engaged to Konecki.