Lord Frederick Windsor was by the actress' side as she delivered their daughter, named Maud Elizabeth Daphne Marina, in a hospital in Los Angeles, where the couple are based.
In a statement confirming the happy news, Prince and Princess Michael said, "Lady Frederick Windsor was yesterday safely delivered of a baby girl at Ronald Reagan UCLA Medical Centre in Los Angeles.
"The weight of the baby was 7lb. Lord Frederick Windsor was present at the birth.
"Prince and Princess Michael of Kent's family and Sophie's family have been informed and are delighted with the news."
The Queen, who is a cousin of Prince Michael, has been told of the arrival of the little girl who is 42nd in line to the throne.
Lord Freddie, 34, and Sophie, 33, married in September 2009 at Hampton Court.
Cambridge-educated Sophie met the Old Etonian banker on a night out with friends in 2007 and got engaged on Valentine's Day in 2009.
They live in Los Angeles where he works as a financial advisor and Sophie, who has starred in TV sitcoms Peep Show and Two and a Half Men, is pursuing her acting career.
The talented actress is the daughter of children's book author Cindy Black and the half-sister of TV star Claudia Winkleman.