The performance expected from a galaxy survey to be carried out with GAIA, the GAIA Galaxy Survey, is outlined. From a statistical model of galaxy number density, size and surface brightness distribution, and from detailed numerical simulations based on real images, it is conservatively estimated that GAIA would be able to detect and observe about 3 million galaxies brighter than V ≅ 17 and to provide multi-colour and multi-epoch broad-band photometry of these with an end-of-mission angular resolution of ≅0.35 arcsec and a photometric accuracy of ≅0.2 mag/arsec2 at μV=20 mag/arsec2. The substantial scientific case for performing such a survey and the additional efforts required in terms of mission preparation, operations and telemetry are also discussed.