A source close to the Cambridges said: "The Duke and Duchess are absolutely overjoyed by the happy news."
It comes after a family source told HELLO! that Pippa gave birth on Monday 15 March at around 4:22am.
The couple have decided to name their daughter, who weighed 6lbs 7oz, Grace Elizabeth Jane. "Mother and baby are doing well," the source said. "She's perfect, everyone is overjoyed at such a happy arrival."
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Pippa was spotted enjoying a stroll around London on Saturday, wearing a casual green coat that skimmed over her baby bump.
She and James, who married in May 2017, are already parents to two-year-old Arthur.
The new arrival is the fifth grandchild for Pippa's parents Carole and Michael Middleton, and another cousin for Prince William and Kate's three children, Prince George, Princess Charlotte and Prince Louis, to play with.
Pippa and James married in May 2017
The birth of Pippa's daughter is even more special as the baby arrived just one day after Mother's Day.
The palace also posted a picture of the cake the children made for mum Kate, as well as a childhood snap of the Duchess with her mum Carole.
Kensington Palace shared a rare childhood photo of Kate with mum Carole
Businesswoman Carole confirmed Pippa's second pregnancy in the April issue of Good Housekeeping.
The Party Pieces founder was asked about her hopes for 2021 in the cover interview.
Carole said: "I hope we see COVID-19 coming under control so all of us can have a more normal life. Hopefully this year, more people will get to celebrate their special occasions with friends and family.
"I also hope to see more of my family than I could last year, including, of course, my new grandchild."
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