Barça already had an option to sign Mina in summer for a fixed price of €9m, but an agreement to bring that forward by several months has long been expected.
The Catalan giants have confirmed that they have paid €11.8m for the Colombian centre-back, with 23-year-old Mina agreeing a long-term contract until June 2023.
His buyout clause has been set at €100m.
Barça see great 'potential' in Mina and in their welcome statement the club speak highly of the 'physical presence' and 'aerial prowess' they expect the player to bring to the defensive unit.
Having started out at Deportivo Pasto at the age of 19, Mina got his big break at Independiente Santa Fe before leaving his native Colombia and heading to Brazil with Palmeiras in 2016.
He helped the team win Brazil's notorious competitive Campeonato Brasileiro for the first time since 1994 in his debut season. Mina had earlier tasted success in Colombia with Santa Fe after winning the 2015 national title as well as that year's Copa Sudamericana.
It is expected that Mina's arrival at Camp Nou could well seal Javier Mascherano's expected exit to China after seven-and-a-half years of service.