The Brazilian had spent the last two seasons on loan in Istanbul and the two clubs have now confirmed that the deal has been made permanent for a fee of 3.75million euros.
"Felipe Melo will continue to wear the shirt of Galatasaray," said a statement on www.juventus.com.
"After playing the last two seasons in Istanbul, the Brazilian midfielder is now the Turkish club's player.
"Juventus have reached an agreement with Galatasaray for the outright sale of Felipe Melo for a price of 3.75 million euros."
Melo has signed a three-year deal in the Turkish capital.