The Duke of Cambridge, who became a father just three weeks ago with the birth of his son George, was making a visit to his Royal Air Force (RAF) base at Anglesey in Wales.
It was one of Prince William's first public engagements since the birth of Prince George in London on July 22.
"I have to say that I thought search and rescue duties over Snowdonia were physically and mentally demanding, but looking after a three-week-old baby is up there... He's pretty loud but of course very good-looking," he said during a speech at the Anglesey's annual agricultural show.
"I know that I speak for Catherine (Middleton) when I say that I have never in my life known somewhere as beautiful and as welcoming as Anglesey.
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"I know that both of us will miss it terribly when my search and rescue tour of duty comes to an end next month and we have to move elsewhere," he said.
"From the bottom of my heart, thank you for making my wife and me so welcome when we arrived here, as you do thousands of visitors each year. Catherine and I look forward to returning again and again over the coming years with our family," he said.
The visit is expected to mark the duke's farewell to Anglesey at the end of his tour of duty at RAF Valley.