We model for the first time the complete orbital evolution of a pair of supermassive black holes (SMBHs) in a 1:10 galaxy merger of two disk-dominated gas-rich galaxies, from the stage prior to the formation of the binary up to the onset of gravitational wave (GW) emission when the binary separation has shrunk to 1 mpc. The high-resolution smoothed particle hydrodynamics (SPH) simulations used for the first phase of the evolution include star formation, accretion onto the SMBHs as well as feedback from supernovae explosions, and radiative heating from the SMBHs themselves. Using the direct N-body code phi-GPU, we evolve the system further without including the effect of gas, which in the mean time has been mostly consumed by star formation. We start at the time when the separation between two SMBHs is ~700 pc and the two black holes are still embedded in their galaxy cusps. We use three million particles to study the formation and evolution of the SMBH binary until it becomes hard. After a hard binary is formed, we reduce (reselect) the particles to 1.15 million and follow the subsequent shrinking of the SMBH binary due to three-body encounters with the stars. We find approximately constant hardening rates and that the SMBH binary rapidly develops a high eccentricity. Similar hardening rates and eccentricity values were reported in earlier studies of SMBH binary evolution in the merging of dissipationless spherical galaxy models. The estimated coalescence time is ~5.5 Gyr, significantly smaller than a Hubble time. We discuss why this timescale should be regarded as an upper limit. Since 1:10 mergers are among the most common interaction events for galaxies at all cosmic epochs, we argue that several SMBH binaries should be detected with currently planned space-borne GW interferometers, whose sensitivity will be especially high for SMBHs in the mass range considered here.