We study the dynamical evolution of stars and gas close to the centre of the Milky Way. Any plausible means of forming the young stars observed at the Galactic Centre leaves behind a residual gas disc at ∼0.01 pc radii. We show that the combined effects of viscous accretion and gravitational interactions with stars do not remove the residual gas efficiently, and that a substantial gas disc, interior to the stellar disc, persists for >10 Myr after the stars form. Since no such disc is currently seen at the Galactic Centre, we argue that it has been accreted by the supermassive black hole. This scenario offers an attractive connection between nuclear star formation and black hole feeding, and we suggest that the ‘missing’ gas may have been used to power Sgr A*.