The project stalled three months ago after just getting underway when contractor Shaylor suddenly collapsed.
Initial works, including the removal of asbestos, were completed but the main part of the job, has yet to get underway.
Leader of the City Council Ian Brookfield said the decision to bring Willmott Dixon Interiors on board would get the project back on track to be delivered on time and within budget.
He added that an official application for Willmott Dixon to take on the job was expected to go to planning bosses next week.
The scheme to revamp the 80-year-old Civic and Wulfrun halls will see the combined capacity at both halls increased to 4,600 standing and 3,130 seating.
The Civic Hall is currently scheduled to reopen in 2021 and it is hoped the Grand Slam of Darts will be one of the first events at the venue when it reopens.