The council has published an OJEU notice inviting firms to tender and expects to draw-up a shortlist of between five and 10 contractors.
Interested companies have until June 21 to get their tenders in before the shortlist is announced in August.
Construction of the new four-storey block is expected to take 60-70 weeks following the anticipated award by the end of December.
The single-stage design and build job is due for completion by May 2012.
West Dorset is working with Simons Developments on the scheme with architect Carey Jones, structural and civil engineers BWB and services engineer Hannan Associates.
Details of public and private work opportunities are available free every day on this site. Click here to view the latest schemes up for grabs.