According to the statements released by Barcelona and Juventus, the deal is worth €72 million (£66m/$81m) plus up to €10m in bonuses.
Juve’s Miralem Pjanic will also move in the other direction, although both players will conclude the 2019-20 campaign at their current clubs.
Initially, Arthur insisted he was happy in Barcelona and not ready to move but he changed his decision on Thursday, which paved the way for the deal to be completed.
“FC Barcelona and Juventus FC have reached an agreement for the transfer of player Arthur Melo. The Italian club will pay €72 million, plus €10 million in variables,” the club confirmed in a statement published on Monday.
Reacting to the development, Barcelona head coach Setien, who included the player in the squad to face Atletico Madrid on Tuesday, said: “I have no doubt about his greatest commitment until the last day he is here.
“He is a player until the last game of the season. He has committed himself to being one more [member of the squad].
“He will dedicate himself fully to achieving the objectives. I have no doubts about him.
“He has said that he will be in the best conditions. I’ll decide if he plays more or less.”
Arthur joined the club from Gremio in 2018 and has turned out 72 times in total for the Camp Nou side, including 28 appearances in all competitions this season, during which he has scored all four of his goals for the club.