We describe in detail our observations of a new mode of phase separation process which occurs during the annealing of amorphous Al-Ge alloys. This process is characterized by the growth of circular colonies, linear in time at fixed temperature, and with an activated growth velocity. The colonies consist of a branching Ge polycrystalline core and of a nearly monocrystalline Al outer rim, with deep fjords extending into the interior of the colony. The spatial distribution of the constituents is characterized by a length scale, roughly equal to the thickness of the Al rim, that decays exponentially as the temperature is increased. The Al rim contains a fairly large concentration of Ge atoms. The measured values of the characteristic length scale and of the growth velocity appear to be consistent with the assumption that the growth is controlled by the diffusion of Ge atoms from the surrounding amorphous Al-Ge matrix through the crystalline Al rim.