We report the discovery of an eclipsing low-mass X-ray binary at the center of the 3FGL error ellipse of the unassociated Fermi/Large Area Telescope {gamma}-ray source 3FGL J0427.9-6704. Photometry from OGLE and the SMARTS 1.3m telescope and spectroscopy from the SOAR telescope have allowed us to classify the system as an eclipsing low-mass X-ray binary (P=8.8hr) with a main-sequence donor and a neutron-star accretor. Broad double-peaked H and He emission lines suggest the ongoing presence of an accretion disk. Remarkably, the system shows separate sets of absorption lines associated with the accretion disk and the secondary, and we use their radial velocities to find evidence for a massive (~1.8-1.9M_{sun}_) neutron-star primary. In addition to a total X-ray eclipse with a duration of ~2200s observed with NuSTAR, the X-ray light curve also shows properties similar to those observed among known transitional millisecond pulsars: short-term variability, a hard power-law spectrum ({Gamma}~1.7), and a comparable 0.5-10keV luminosity (~2.4x10^33^erg/s). We find tentative evidence for a partial (~60%) {gamma}-ray eclipse at the same phase as the X-ray eclipse, suggesting the {gamma}-ray emission may not be confined to the immediate region of the compact object. The favorable inclination of this binary is promising for future efforts to determine the origin of {gamma}-rays among accreting neutron stars.