The Spanish champions will pay an extra 9 million in bonuses in the six-year deal struck with the Brazilians for the midfielder, full name Arthur Henrique Ramos de Oliveira Melo.
A release clause has been set at 400 million, the Catalan giants said overnight Monday on their website.
Barca had inked a pre-accord for the 21-year-old in March.
Arthur was a star turn with Gremio as the Porto Alegre side lifted the Copa Libertadores, and was named as man of the match in their final win over Argentina's Lanus.
At the Nou Camp he will notably link up with compatriot Philippe Coutinho.