The 22-year-old completed his switch to Oakwell ahead of transfer deadline day and has signed a three-year contract with the South Yorkshire club.
Darlington-born Jackson, a product of the club's Academy who has represented England at youth levels, enjoyed a highly successful loan spell at Hartlepool United last season, making 33 appearances.
The centre-back featured with Aitor Karanka's Boro squad during pre-season and joined the summer training camp in Marbella.
Everyone at MFC would like to wish Adam all the best for the future.
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