The 21-year-old had initially been due to arrive at Camp Nou at the end of the year upon the conclusion of the Campeonato Brasileiro Serie A campaign.
Arthur, yet to be capped by Brazil at senior level, has been handed a six-year contract at Camp Nou which includes an astronomical buyout clause of £355m.
A statement on the La Liga champions' official website read: "FC Barcelona and Gremio de Porto Alegre have agreed the transfer of the player Arthur Henrique Ramos de Oliveira Melo, following the agreement the two clubs reached in March.
"The agreement will see the Brazilian player immediately join the squad. The cost of the transfer is off €31m with €9m in variables. The player will sign a contract with the club for the following six seasons, until the end of the 2023/24 campaign; his buyout clause is to be set at €400m.
"The schedule for the presentation of Arthur Henrique Ramos de Oliveira Melo as a new first team player will be announced shortly."
Arthur admits that he is "fulfilling a dream" by signing for the Catalan giants.