Armie and Elizabeth Chambers Hammer Name Son Ford Douglas Armand

Armie Hammer's Social Network just grew by one!

The actor and his wife, Chief Correspondent of the Human Rights Foundation Elizabeth Chambers Hammer, welcomed their second child on Sunday, Jan. 15, in Los Angeles, his rep confirms to PEOPLE exclusively. The baby boy joins the couple's daughter Harper, 2.

"Emerging from his sister's smooches to say hello," Chambers Hammer captioned a photo of her children, revealing her newborn son's name. "Welcome to the world, Ford Douglas Armand Hammer!"

"Both Mom and baby are doing great," the rep tells PEOPLE.

"Armie and Elizabeth are overjoyed with their growing family and daughter Harper is already embracing the role of big sister," adds a source close to the couple.

On Wednesday, Chambers took to Instagram to share a family photo from the hospital.

"Beyond grateful for the prayers and for this ultimate blessing. We couldn't be more in love," she captioned the sweet snap of the proud big sister hugging her baby brother.

The new mom of two followed up the photo with a second shot of Harper sporting her big sister dress. "Homeward bound and just a little excited," she wrote.

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Hammer, 30, and his journalist wife, 34 — who stars as a judge on Celebrity Cupcake Wars and recently opened a second Dallas location of her bakery BIRD — have been married since May 2010.

The couple announced they were expecting baby number two on Instagram in September, with the Free Fire star revealing that their newest addition would be a boy on Jimmy Kimmel Live! later that month.

"We're thinking Jack," Hammer joked about their name ideas, adding, "[Or] Michael Charles, and he can just go by M.C."

"Are you really thinking about that?" asked Kimmel, who is currently expecting his second child with wife Molly McNearney.

"I think I just gave away our baby name?" he replied. "I'm in so much trouble!"

Hammer added that his daughter was really excited about her upcoming new role as big sister.

"She loves it," he said. "I mean, she seems to love it now. If you go, 'Where's the baby?' she'll rub Mama's belly and kiss her stomach. It's really cute."

But the actor knows that as sweet as she is, bringing home a newborn won't be easy for his little girl.

"I think when the new baby comes out, it's going to be a serious adjustment period," he added.

Hammer — who will next be seen in Call Me By Your Name and Hotel Mumbai — has kept a refreshing perspective on parenting.

"I don't know if our daughter's easy or if we've just been winging it the right way, but we haven't had too much of a plan and it's all kind of gone all right," he told Jimmy Kimmel.

"I guess the only reason we're all here is because raising babies works. I think people — especially new parents, at least I did — underestimate the resilience of human survivalism. Like, this thing is going to survive. That's the only reason it's here."

• With reporting by JULIE JORDAN