PEOPLE exclusively announced Wednesday that Scarlett Johansson and Colin Jost welcomed a baby

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Colin Jost is loving fatherhood.

The Saturday Night Live star, 39, and wife Scarlett Johansson recently welcomed their first baby together, Johansson's rep Marcel Pariseau exclusively confirmed to PEOPLE Wednesday.

Later, on Instagram, Jost shared a quick statement, revealing that their baby boy is named Cosmo — and joking that people can reach out to SNL costar Michael Che for more.

"Ok ok we had a baby. His name is Cosmo. We love him very much," writes Jost. "Privacy would be greatly appreciated. For all inquires please contact our publicist @chethinks."

He also added the hashtags: "#wegotawaywithitforalongtime" "#nokidspolicy" and "#weregoingtodisneyworld."

Johansson is also mom to daughter RoseDorothy, who turns 7 years old this fall. The Sing 2 actress and Jost wed in an intimate ceremony back in October 2020. The pair got engaged in May 2019 after two years of dating.

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In October 2019, Johansson told Ellen DeGeneres that Jost "killed it" with his "surprising" proposal, which included "a whole James Bond situation."

For more on Scarlett Johansson and Colin Jost's baby news and other top stories, listen below to our daily podcast PEOPLE Every Day.

"He's got a lot behind that news desk he's hiding," she joked, referring to Jost's job as "Weekend Update" co-host on SNL.

"He's very charming and very thoughtful and romantic. But yeah, I was surprised. Even if you kind of imagine what that moment's gonna be like, it's still a beautiful moment."