The couple was "nervous, and it took them a while before they could relax and fully enjoy this pregnancy"

Meghan Markle and Prince Harry are "overjoyed" for their rainbow baby.

The couple announced on Sunday that they are expecting their second child — months after revealing that they suffered a miscarriage last July.

Meghan, 39, and Prince Harry, 36, were "hopeful that they would get pregnant again. And they were overjoyed that it happened so quickly," a close source to the couple tells PEOPLE in this week's issue.

Still, after the "devastating" miscarriage, "they were both nervous, and it took them a while before they could relax and fully enjoy this pregnancy," the source adds.

Harry and Meghan Markle announce pregnancy
Prince Harry and Meghan Markle's pregnancy reveal on Feb. 14
| Credit: Misan Harriman
meghan and harry
Meghan Markle and Prince Harry attending the Commonwealth Day Service in the U.K. in March 2020.
| Credit: Shutterstock

The arrival of Meghan and Prince Harry's second child will make Archie Harrison, who will turn 2 on May 6, a big brother.

"They always wanted for Archie to have a sibling close in age," the source tells PEOPLE.

In fact, the couple's firstborn was a centerpiece in their new baby announcement. "We can confirm that Archie is going to be a big brother. The Duke and Duchess of Sussex are overjoyed to be expecting their second child," a spokesperson for the couple told PEOPLE.

Prince Harry, Duke of Sussex, Meghan, Duchess of Sussex and their baby son Archie Mountbatten-Windsor
Prince Harry, Archie and Meghan Markle
| Credit: Pool/Samir Hussein/WireImage

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Meghan and Harry — who will sit down with Oprah Winfrey for an "intimate conversation" airing on March 7 on CBS — announced their pregnancy news with a stunning black-and-white photo that was taken remotely via iPad by their friend and longtime photographer, Misan Harriman. A smiling (and barefoot!) Harry looks lovingly at Meghan, who cradles her baby bump as she rests in his lap.

"They are besotted parents, and they can't wait to share that even more with a second child," a royal insider tells PEOPLE. "They are both so happy to have their little family."