In a statement, Buckingham Palace said: "The Duke of Sussex acknowledges and welcomes the formal apology".
A court statement read out by Harry's solicitor added: "On 9 January 2019 Splash chartered a helicopter for the goal of taking photographs and recording video footage of The Duke's private home in Oxfordshire".
"The property had been chosen by The Duke for himself and his wife given the high level of privacy it afforded given its position in a secluded area surrounded by private farmland away from any areas to which photographers have access", started the statement.
"The syndication and publication of the photographs very seriously undermined the safety and security of the Duke and the home to the extent that they are no longer able to live at the property", the lawyer said in a court statement.
Splash says it "recognized that this situation represents an error of judgment" and promised it would not happen again.
Previously, the royal couple lived in a lavish country home in Britain's Cotswolds-a picturesque area of south-central England.
And it is reported the Sussexes' new household at Buckingham Palace is actually about having tighter control over the couple.
The Duke of Sussex instructed his lawyers to act after the picture agency Splash hired a helicopter to take photographs of the house which Harry and Meghan were renting in Oxfordshire.
Angela, who has interviewed Harry in the past during her time as a royal reporter, told the Daily Mail: "The intention is to ensure what they do is co-ordinated with the rest of the household so you don't get an Independent State of Frogmore. The Duke and Duchess's official office will continue to be based at Kensington Palace".
There were reports earlier this week that Meghan and Harry had invited Gayle into their home for a chat just before Meghan went into labour, but a source has since told The Daily Beast that's not true.
Royal correspondent Omid Scobie reports that Splash News agency took pictures from a helicopter on 9 January. Markle's mother, Doria has reportedly been helping the new mother with baby Archie instead.