Pippa Middleton welcomed her second child with her husband James Matthews on Monday, March 15, and the Cambridges are "overjoyed" with the news.

Pippa, the younger sister of Duchess Kate Middleton, gave birth to a daughter, weighing 6lbs 7oz, at 4:22 am on Monday. The baby has been named Grace Elizabeth Jane Matthews. Grace's middle names are a nod to her grandmothers, Carole Elizabeth Middleton and Jane Matthews. Elizabeth is also the middle name of Kate, whose full name is Catherine Elizabeth Middleton.

A source close to Kate and Prince William told Hello! magazine that both the mother and the baby are "doing well," and that "the Duke and Duchess are absolutely overjoyed by the happy news."

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

Grace is the second child for Pippa and James, who are also parents to a two-year-old son, Arthur Michael William Matthews. Arthur's middle name Michael is a tribute to his maternal grandfather Michael Middleton and to his father's younger brother Michael Matthews who died while climbing Mount Everest in 1999. Arthur shares his first name with his royal cousin Prince Louis, who uses it as a middle name.

His newly-born sister is the fifth grandchild for Pippa and Kate's parents Carole Middleton and Michael Middleton, who are also grandparents to England's future King Prince George.

The birth of Pippa's daughter comes just two weeks after Carole confirmed it to the public. "I hope to see more of my family than I could last year, including, of course, my new grandchild," the businesswoman told Good Housekeeping UK earlier this month.

Meanwhile, there have been reports that Kate is also trying for another baby. The future Queen consort, who will celebrate a decade of her marriage with Prince William next month, is already mother to Prince George, seven, Princess Charlotte, five, and Prince Louis, two. "Having four children was always part of Kate's plan...After taking some time to think about it, he's (William) on the same page and is excited about the future," a source told Us Weekly.

Kate and Pippa
Britain's Catherine, Duchess of Cambridge sits with her sister Pippa Middleton Reuters