Kate Middleton and Prince William 'Overjoyed' by Pippa Middleton's Baby News

Pippa and James Matthews welcomed their second child, a baby girl, on Monday

Kate Middleton and Prince William are an aunt and uncle again!

The royal couple are "overjoyed" by the arrival of Kate's younger sister Pippa Matthews' second child, a little girl named Grace Elizabeth Jane.

A source close to the couple tells PEOPLE, "The Duke and Duchess are absolutely overjoyed by the happy news."

Grace Elizabeth Jane was born around 4.22 a.m. this morning, weighing 6lbs, 7oz. Mother and baby are doing well, a family source tells PEOPLE.

"She's perfect, everyone is overjoyed at such a happy arrival," the source adds.

Kate Middleton And Pippa Middleton
Pippa Middleton, Kate Middleton and Prince William in 2011. KIRSTY WIGGLESWORTH/AFP via Getty

Pippa and James Matthews' newborn daughter shares a middle name with her aunt Kate, who was born Catherine Elizabeth Middleton. (She also shares the name of Kate's grandmother-in-law, Queen Elizabeth!)

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Kate Middleton And Pippa Middleton
Kate Middleton And Pippa Matthews. Karwai Tang/Getty

Pippa, 37, and James, 45, welcomed their first child in Oct. 2018. Pippa followed in the footsteps of her royal sister by giving birth in the Lindo Wing of St. Mary's Hospital — the same place where Kate and Prince William welcomed all three of their children.

Their baby girl is the fifth grandchild for Carole and Michael Middleton and a little cousin for Prince George, 7, Princess Charlotte, 5, and Prince Louis, 2.

James Matthews, Pippa Middleton and Kate Middleton. Kirsty Wigglesworth/AP

Carole, who was recognized by Kate and Prince William while celebrating Mother's Day in the U.K. over the weekend, recently described herself as an active influence in her grandchildren's lives, something she hopes will continue.

"I want to run down the hills, climb the trees and go through the tunnel at the playground," she says. "I cook with them, I muck around dancing, we go on bike rides."

Discussing life after lockdown and her hopes for the year ahead, the grandmother said: "I also hope to see more of my family than I could last year, including, of course, my new grandchild."

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There's also more excitement in the Middleton family — Kate and Pippa's brother James got engaged to girlfriend Alizee Thevenet last year. They decided to postpone their wedding scheduled for May 2020 due to the ongoing coronavirus pandemic.

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