We investigate numerically the evolution of black hole binaries embedded within geometrically thin gas discs. We find that the binary decays at a rate which is in good agreement with analytical estimates. Scaling our results, we calculate the maximum rate of binary decay that is possible without fragmentation occurring within the surrounding gas disc. We find that decay due to gas discs can be more efficient than that due to stellar scattering for separations below a∼0.01-0.1 pc. The minimum merger time scale is shorter than the Hubble time for M<107 M⊙. This implies that gas discs could commonly attend relatively low mass black hole mergers, and that a significant population of binaries might exist at separations of a few hundredths of a pc. For more massive binaries, scattering of stars formed within a fragmenting gas disc could act as a significant additional sink of binary angular momentum.