We present the results of the first multi-scale N-body+hydrodynamical simulations of merging galaxies containing central supermassive black holes (SMBHs) and having a spatial resolution of only a few parsecs. Strong gas inflows associated with equal-mass mergers produce non-axisymmetric nuclear disks with masses of order 109 M ⊙, resolved by about 106 SPH particles. Such disks have sizes of several hundred parsecs but most of their mass is concentrated within less than 50 pc. We find that a close pair of supermassive black holes forms after the merger, and their relative distance then shrinks further owing to dynamical friction against the predominantly gaseous background. The orbits of the two black holes decay down to the minimum resolvable scale in a few million years after the merger for an ambient gas temperature and density typical of a region undergoing a starburst. The conditions necessary for the eventual coalescence of the two holes as a result of gravitational radiation emission appear to arise naturally from the merging of two equal-mass galaxies whose structure and orbits are consistent with the predictions of the LCDM model. Our findings have important implications for planned gravitational wave detection experiments such as LISA.