We demonstrate liquid-metal-mediated homoepitaxial crystal growth of Ge on Ge(111) at temperatures in the range of 400-450°C. Crystal growth proceeds by diffusion of Ge through a liquid layer, followed by precipitation onto the substrate by the vapor-liquid-solid mechanism. The liquid-metal phase at the interface is a Au-Ge alloy formed by initial deposition of a thin Au layer above the eutectic temperature. Ge vapor is provided by a molecular-beam evaporator. The resulting films revealed high-crystalline quality by in situ high-energy ion scattering and channeling analysis and ex situ by cross-sectional transmission electron microscopy.