The actor celebrated his 49th birthday on Sunday alongside Lopez and his three children

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Ben Affleck's birthday celebration was a quiet but close affair.

On the evening of his 49th birthday on Sunday, Affleck shared a "quiet celebration" alongside Jennifer Lopez and her 13-year-old twins, Max and Emme, a source close to Lopez tells PEOPLE in this week's issue.

"Jennifer had a cake for him. Ben is not one for big celebrations, so he thought it was perfect," says the source. "It was exactly what he wanted."

Earlier on Sunday, the Oscar-winning director and actor enjoy some time alone with his kids Violet, 15, Seraphina, 12 and Samuel, 9, whom he shares with ex-wife Jennifer Garner.

For more on Jennifer Lopez and Ben Affleck's rekindled romance and other top stories, listen below to our daily podcast PEOPLE Every Day.

Before meeting with Affleck, Lopez and Emme were spotted at producer Jennifer Klein's home for her annual Day of Indulgence event where the singer and the teenager were spotted picking out jewelry for Violet and Seraphina. 

"When she stopped by the Made by Mary jewelry station, she and Emme picked out several 'birth flower necklaces,' including two for Ben's daughters," a guest tells PEOPLE, noting that Lopez said she planned to gift them to Violet and Seraphina at Affleck's birthday gathering.

The couple, who recently vacationed in St. Tropez for Lopez's birthday, are getting serious as Lopez plots a move from Miami to Los Angeles, where Affleck is based. An insider previously told PEOPLE that Emme and Max are "onboard with starting fresh in Los Angeles."

"They are slowly getting to know Ben," the source added. "Everything seems to be running smoothly. It's very obvious that Jennifer is serious about Ben. She hasn't looked this happy for a long time."

Jennifer Lopez and Ben Affleck
Credit: Elder Ordonez/SplashNews.com

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Following their respective splits from Alex Rodriguez, 45, and Ana de Armas, 33, Lopez and Affleck rekindled their romance in late April. They originally began dating in July 2002 after meeting on the set of their movie Gigli. They got engaged that November before postponing their September 2003 wedding just days before the original date and ultimately called off their engagement in January 2004.

A source told PEOPLE in May that their relationship is different this time around. "Time has passed, and they are each in a different place with children," the insider said.

For more about Ben Affleck and Jennifer Lopez, pick up the latest issue of PEOPLE, on newsstands Friday.