The 21 year-old moves to Juve in a permanent transfer.
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Costa has signed a contract to 2029 with the Bianconeri.
Juve announced they will pay Vitoria €12.5m over four installments, plus bonuses not exceeding €2.5m.
With Bremer and Juan Cabal out for the season due to injury, Juve have brought in Costa as readymade cover.