Architect Snell Associates and consulting engineer Arup will be working on the project, which will include extending galleries and project spaces, and adding an education centre and a 60-seat studio.
Work is due to be completed in May 2005.
An award of £7.5m from the National Lottery's Arts Council in September 2001 enabled Arnolfini to buy Bush House from property holding company Ivory Gate (Gibraltar).
Further funding for the £12m project has been provided by Bristol council, the South West Regional Development Agency and local businesses and individuals.
Sir Nicholas Serota, director of Tate, has championed the refurbishment project.
He said: "The designs will enable the organisation to take its place as one of a handful of the most significant contemporary cultural centres in Europe."
The project also includes a new resource area with a library and archive and the refurbishment of the Arnolfini's bookshop and cafe.
The building will be closed from 29 September and will reopen to the public in spring 2005.