Arthur, 21, has signed a six-year contract at the Nou Camp after the clubs reached an agreement in March.
The deal, which includes a 400m euro (£451m) buy-out clause, will initially cost Barcelona 31m euros (£27.5m).
It comes a day after Brazil midfielder Paulinho left La Liga’s champions to rejoin Chinese club Guangzhou Evergrande on loan.
He has been called up to the Brazil team in the past but has yet to play for them, and did not make their World Cup squad.
The midfielder helped Gremio win Brazil’s equivalent of the FA Cup, the Copa do Brasil, in 2016 and South America’s Champions League – the Copa Libertadores – in 2017.
In total, he scored six goals in 66 appearances for Gremio.