The pair had reportedly opted not to learn the baby’s sex before delivery.
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In an appearance on "Jimmy Kimmel Live!" last year, Schwartzman joked he was growing his signature ‘stache to “embarrass my unborn child.”
“I decided to grow out this mustache, so when we take photos…my boy or girl, whatever it is...they will be embarrassed,” he said back in August.
Cunningham, a fashion designer, and Schwartzman wed July 11, 2009 at their home in San Fernando, Calif.