The 32-year-old left-back returns to his native Brazil after 12 years in Europe, split evenly between a six-year spell at Valencia and his time at Anfield.
Aurelio was troubled by injuries while at Liverpool and was allowed to leave at the end of the 2009/10 season before rejoining the club that August after Roy Hodgson became manager.
Further injury problems followed, however, and he barely featured in the last 12 months under Kenny Dalglish, making just two Premier League appearances in the 2011/12 campaign.
His new club announced the acquisition on their official website, stating: "Gremio confirm agreement with the left-back Fabio Aurelio Rodrigues. The player's contract extends to December 31, 2013.
"The player, developed at Sao Paulo, has been with Liverpool in England since 2006 and also served for six years at Valencia in Spain."
Liverpool's Brazilian midfielder Lucas Leiva wished Aurelio well, writing on Twitter: "Gremio signed Fabio Aurelio. What a great news."