Eugenie, a granddaughter of Queen Elizabeth II, and her husband, Jack Brooksbank, welcomed their son at London's Portland Hospital on Tuesday morning.
Eugenie's parents are Prince Andrew and Sarah, Duchess of York, The princess' baby, who weighed 8 pounds, 1 ounce, is her and Brooksbank's first child and the queen's ninth great-grandchild.
"Her Royal Highness and her child are both doing well," the statement said.
The palace said that Elizabeth and her husband, Prince Philip, were "delighted with the news."
Eugenie married Brooksbank, a businessman, in October 2018 at St. George's Chapel at Windsor Castle.
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