The four-storey project, which includes library and underground parking, was also chased by Kier, Galliford Try, Sir Robert McAlpine, Vinci and Wates.
Council partner developer Simons Group is building the offices as part of a bigger £60m mixed scheme involving 20 shops, including a Waitrose store, flats, hotel and an underground car park with 485 parking spaces.
Balfour Beatty’s Bristol-based subsidiary Cowlin will start work on the council building in June.
The second phase, which is being tendered separately, is penciled in to start next year.
The development comes after 30 years of failed schemes and debates over the prime site which is covered by car parks.
Cowlin divisional managing director Neil Sherreard said: “During the construction programme, we estimate we will be employing up to 100 people on site, many of whom will be from the local area.
“Cowlin has a proud track record of delivering schemes of this nature and we look forward to completing a development that the people of Dorchester will benefit from for years to come.“
Architect Carey Jones, structural engineer BWB and services engineer Hannan Associate, appointed by Simons, have now been novated to the contractor.