The 24 year-old Ivorian striker will join the club for a record £12million after previously having an agreement earlier in the week.
The previous record was a fee closer to £7million when they brought in Pablo Hernandez from Valencia last summer.
"This move has been talked about for quite some time, so I am delighted it is all completed," he told the club's official website.
"Now it is important that I focus my mind on the challenge ahead because I hope to make the Swansea fans very happy."
Bony scored an amazing 31 goals in 30 appearances for Vitesse in the Dutch league last season and the hope is that Swansea have found a reliable goal scorer to help cement their stay in the top flight with many goals last season only coming from Michu.