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The actor and author's wife, Ryan Shawhughes, gave birth to Clementine Jane Hawke on Friday in New York City.
"They are thrilled," Hawke's rep, Mara Buxbaum, said when news of the pregnancy was first announced.
"I expect that they should be very good parents," a close family member told PEOPLE Wednesday. "They both love children, and they love each other ... I know they are thrilled."
Hawke, 37, and Shawhughes – former nanny to Hawke's son and daughter with Uma Thurman – married in June in a small ceremony in New York City.
– Michael Y. Park, Karen Snyder and Julie Jordan
JOSH & FERGIE: ROCKED BY SCANDAL
Did he cheat with a stripper?
Married less than a year, the couple denies an Atlanta woman’s claims that she and Josh had a fling
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