The scheme will become a key landmark in the residential-led redevelopment east of the city centre.
Initial estimates for the project, which was unveiled in November 2005, put the cost of the new office at £27.6 million.
However, after detailed examination of the figures, it emerged that the cost had risen to £31.1million, and the latest estimates place the eventual budget at nearer £35 million.
RMJM will work alongside mechanical and electrical engineer Gifford on the project, which is expected to start in 2008.
The authority hopes to move its staff in to the new building in 2010.