Along with Si, Ge in β-Ga$_2$O$_3$ is also a promising n-type dopant. To achieve low contact resistance for device applications, n-type layers with carrier concentration above 10$^{19}$ cm$^{-3}$ are required. Homoepitaxial (010) β-Ga$_2$O$_3$ films grown by MOCVD were implanted with Ge to form box profiles of various conditions. Lattice recovery was investigated using X-ray diffraction (XRD) for anneals from 100°C to 1050°C. The $\gamma$-phase was observed for all samples and phase recovery was shown to be achieved through a common oxygen sub-lattice. For low-damage implants, annealing under research plus N$_2$ between 950°C and 1000°C for 5 to 10 minutes results in sheet resistance of 600-700 Ω/∎ with mobilities of 60-70 cm$^2$/V•s, and up to 40% Ge activation. Ge diffusion during annealing was determined by secondary ion mass spectrometry (SIMS). For anneals at temperature for activation, a Ge cluster peak with concentrations above the initial implant were observed, with the peak occurring near the estimated maximum damage depth. The measured active Ge concentration is shown to correspond to the fraction of Ge outside this clustering peak suggesting that peak is electrically inactive. Use of a SiO$_2$ cap during annealing is demonstrated to reduce dopant diffusion at 950°C with no significant impact on dopant activation.