The city's rugby club also plays its home games at the venue. Paisley construction business Barr has previously worked on football stadium projects for Celtic and Southampton as well as building the basketball arena for the London Olympics.
In Bristol the work will take place in three phases over two years with the first seeing the demolition and rebuilding of the 86-year-old Wedlock Stand.
The new structure will include conference facilities, administration offices, a sports bar, cafe and shops.
That will be followed by revamps of the Dolman and Williams stands which will eventually transform Ashton Gate into a 27,000 capacity all seated stadium.
Stephen Lansdown, co-founder of broker and financial services firm Hargreaves Lansdown, owns Bristol City and Bristol Rugby through the Bristol Sport company.
Andrew Billingham, chief executive of Bristol Sport, said: "We want this stadium to be a beacon for sport and entertainment in the South West."