Pippa Middleton Welcomes Baby Girl — and Her Name Has a Special Connection to Aunt Kate!

Pippa Middleton is a mother of two!

The younger sister of Kate Middleton welcomed her second child — a baby girl — with husband James Matthews on Monday, making 2-year-old son Arthur a big brother, PEOPLE has learned.

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.

Pippa and James named their daughter Grace Elizabeth Jane, which means she 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!)

Kate and Prince William are "overjoyed" by the news of Pippa's baby.

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

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.

RELATED: Pippa Middleton Reveals Her 'Saving Grace' While Raising Son Arthur

Mark Cuthbert/UK Press via Getty Pippa Middleton

James Matthews and Pippa Middleton

The new addition is the fifth grandchild for Carole and Michael Middleton and a little cousin for Prince George, Princess Charlotte and Prince Louis.

Pippa and James tied the knot in May 2017 at St. Mark's Church in Englefield. Her royal nephew and niece, Prince George and Princess Charlotte, served in the bridal party.

A source previously told PEOPLE that Pippa and her husband are looking forward to settling down and "having children and leading a quiet life."

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

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Kate's sister documented how she stayed active during her first pregnancy and continued her Waitrose Weekend fitness column by sharing how little Arthur is following in her footsteps by attending a baby gym.

"Now that Arthur is 11 months old and more mobile, I have been trying to come up with different activities to do with him," she wrote in an article released in September 2019.

"I needed to find something more than just park walks in the pram. Our local baby gym has been a saving grace. It's a big space full of fun, soft objects, playmats, stairs, balls, swings, mini trampolines and more to stimulate and physically engage babies and toddlers," she added.

Ian Gavan/Getty

Pippa explained that in addition to free play, there are classes with "structured activities that help promote movement, balance and strength."

"Arthur burns lots of energy in this safe environment and learns many physical skills," she said. "I have also noticed him building his confidence with each visit."

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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 plans this year due to the ongoing coronavirus pandemic.