The cost, estimated at £2.4m in 2018-19, was covered by taxpayers through the Sovereign Grant, but the duke and duchess said they would repay it when they stepped back from royal duties.
Prince Harry's spokesman said he had paid the bill in full by making a contribution to the grant.
The property will remain a UK residence for the duke and his family.
It comes days after the couple announced they had reached a deal with Netflix to make a range of , some of which they may appear in.
A multi-year deal will encompass documentaries, -series, feature films, scripted shows and children's programming.