Feltham will now complete the 210-place Highwood Copse Primary School and become the third contractor involved with the delayed £5m scheme.
West Berkshire Council originally lined-up Kier for the job but after some delays decided to rebid the project because of cost increases.
The retendering process saw the council appoint Dawnus Construction late last summer only for work to stop in March when Dawnus went into administration.
Feltham is now scheduled to complete the job by spring 2020 and was an original bidder for the scheme.
Feltham is now understood to be talking with previous subcontractors with a view to work restarting on site next month.