Meghan, the Duchess of Sussex and wife of Britain's Prince Harry, went into labour in the early hours of Monday, Buckingham Palace said in a statement.
"The Duchess went into labour in the early hours of this morning. The Duke of Sussex (Harry) was by Her Royal Highness' side. An announcement will be made soon," the statement said.
The child, the couple's first, will be seventh-in-line to the British throne.
Queen Elizabeth's grandson Harry, 34, and Meghan, 37, a US-born former actress, married in a lavish ceremony at the monarch's Windsor Castle home in May last year.
The baby will be the eighth great grandchild of the 93-year-old queen, the world's longest-reigning monarch, and could be entitled to hold joint British and American citizenship according to nationality rules.
However, the child will not automatically be a prince or a princess as any such title will have to be bestowed by the queen by issuing what is known as a letter patent.
The couple, officially titled the Duke and Duchess of Sussex, have previously said they intended to keep the plans for the arrival of the new royal private.
After the birth, a notice will be posted on a ceremonial easel on the forecourt of the queen's London home Buckingham Palace, a tradition which dates back at least as far as 1837 when the palace became the sovereign's official residence.
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