Kate’s sister married James Matthews a hedge fund boss, in May 2017 and the couple have a two-year-old son, Arthur.
The couple chose the Queen’s name as a middle name for their daughter.
A family spokesman said: “I’m delighted to confirm, Grace Elizabeth Jane Matthews was born around 4.22am this morning, weighing 6lbs 7oz.
“Mother and baby are doing well.”
The baby will be another cousin for Kate’s children, Prince George, Princess Charlotte and Prince Louis.
Pippa and Kate’s mother Carole Middleton spoke about the birth of her forthcoming grandchild – now her fifth – in an interview with Good Housekeeping magazine.
Commenting on her wishes for the year she said: “I also hope to see more of my family than I could last year, including, of course, my new grandchild.”