The England goalkeeper, 30, has signed a four-year deal with the Magpies.
Pope will return to the Premier League next season after Burnley were relegated to the Championship.
Head coach Eddie Howe said of the signing: "Nick is an exceptional Premier League and international level goalkeeper so I'm very pleased to be adding strong competition to a very important position.
"There has been a lot of interest in him from other clubs this summer, so I'd like to thank our owners and everyone behind the scenes for getting us ahead of the competition. I'm delighted he is joining us as we prepare for the exciting challenge ahead."
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