The 10,000sq ft store is due to shut this week and be re-branded as a Sainsbury's with immediate effect. All staff working in the store will be offered new jobs with Sainsbury's.
The move brings Budgens one step closer to reaching its goal of selling off all of its company-owned stores by the end of the year.
Sainsbury's is also pushing to build a new 25,000sq ft premises close to the Mildenhall store. Plans for the site include the provision of about 175 car-parking places and the relocation of Mildenhall's social and football club.