The Press Complaints Commission (PCC) said it had received a complaint from solicitors after the photograph was published in December.
The magazine printed an apology on its  
          emails page
.
The Duke and Duchess of Cambridge's main home is on Anglesey, where he is a search and rescue pilot at RAF Valley.
The  
          complaint
 was made under clauses in the editors' code of practice relating to privacy and harassment.
The magazine said: "In our issue cover-dated 3 - 9 December 2011, we published a photograph of the Duchess of Cambridge, taken while she was shopping in a store.
"We now accept that we should not have done so, and apologise to her for our actions."
It will be printed in its edition for 26 May - 1 June.
Celebrity magazine Heat has apologised to the Duchess of Cambridge after publishing a photograph of her shopping at a supermarket last year.