The 24-year-old striker became the Swans' record signing earlier on Thursday when he completed his £12million move from Vitesse Arnhem.
Bony completed a medical at the club's pre-season training camp in Holland earlier this week and has signed a four-year deal, which is subject to him being granted a work permit.
The transfer fee more than doubles Swansea's previous record outlay - the £5.5million paid to Valencia for Pablo Hernandez last summer.
Bony, whose nickname in Holland is 'Daddy Cool', is relieved to have finalised the deal and is looking forward to proving himself in the Premier League.
He told the club's official website: "This move has been talked about for quite some time so I am delighted it is all completed.
"Now it is important that I focus my mind on the challenge ahead because I hope to make the Swansea fans very happy.
"I watched Swansea a lot last season on TV. They have some good players and I like the way they play.
"I appreciate the style, which is similar to the Dutch way, and I think being at Swansea will help me develop as a player."
Bony is Swansea's third signing in 10 days, following Alejandro Pozuelo and Jonjo Shelvey in committing to the south Wales team.
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