Harry and Meghan were forced to leave their homes after the photos were published in newspapers, including The Times newspaper, telling CNN a royal source who knew the case well.
In January, Splash News chartered a helicopter to take pictures of the couple's private home in the Cotswolds, England. Buckingham Palace said the couple's safety was compromised.
"The subscription and publication of the photographs seriously damaged the safety of the Duke and his home, as they are no longer able to live on the site," the palace said in a statement.
"The helicopter flew over the house at low altitude, allowing Splash to take pictures of the living room and dining room of the house and directly into the bedroom," he added.
"The Duke of Sussex recognizes and welcomes the official apology presented by Splash News and Picture Agency."
The official residence of Harry and Meghan is Frogmore Cottage in Windsor, near London.
In 2016, Kensington Palace issued a rare statement on behalf of Harry, criticizing the couple's media coverage and noting that "his girlfriend, Meghan Markle, was the focus of a wave of abuse and harassment ".
In a statement sent to the British press agency Press Association, Splash said he "always" acknowledged that this situation was a misjudgment and we have taken steps to ensure that it does not happen again. We apologize to the Duke and Duchess for their distress.
CNN has contacted Splash for a comment.