Ellen Pompeo Describes Watching Her New Daughter Be Conceived (VIDEO)

10/03/2014 AT 09:20 AM EDT

Surprise! Ellen Pompeo is a mom again.

The Grey's Anatomy actress and her husband, Chris Ivery, welcomed their second child, Sienna May Ivery, two months ago, but kept the bundle of joy a secret until Thursday.

After announcing the arrival of Sienna, who was born via surrogate, on Twitter, Pompeo, 44, stopped by Jimmy Kimmel Live! to share more details about the baby, including how she was conceived.

"I was present for the insemination. I held her hand," Pompeo said of the process.

"They make the embryo, and then they insert the embryo into the woman who will carry the child," she said. "This is a incredible thing to do with your life, to give the gift of carrying someone's child, so I am forever grateful and feel very blessed.

"And now we have a gorgeous baby girl."

Kimmel was curious on how the couple kept Sienna a secret for so long. Pompeo reminded the host how she was able to keep her wedding under wraps for five days, even though the mayor of New York City had officiated at the ceremony.

"I've done it before," she said. "The covert thing is kind of fun."

Watch Pompeo's full interview with Kimmel below.

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