The 24-year-old Uruguay international spent last season on loan with the Black Cats and rejoins them on a four-year contract for an undisclosed transfer fee.
Liverpool??paid Nacional ??7m for Coates in 2011 and the defender now becomes Sunderland head coach Dick Advocaat’s first signing as he attempts to rebuild his squad to try and avoid a repeat of the troubles of the last two seasons that saw them flirt with relegation.
The Uruguay defender was signed on loan by compatriot Gus Poyet during his spell in charge on Wearside.
He struggled to make an impact during his early months in the north-east but he came to the fore as the Black Cats launched another ultimately successful safety bid at the end of the campaign.
In total, he made just 13 appearances for the club, but played every minute of the final six games from which Sunderland took nine points to save themselves from relegation.
???Sebastian was an important part of the team under Dick last season and we are delighted that he has joined us on a permanent basis,??? sporting director Lee Congerton told the club???s official website.
???He has Barclays Premier League experience, which is important and, after spending last season on loan with us, he will be able to settle into the team very quickly.???