We have used Low-Energy Electron Microscopy (LEEM) to study the formation and structure of the surface alloy of Bi on the Cu(111) surface. As a function of Bi coverage we find a progression from a dilute surface alloy to a 3×3–R30° structure, which coexists with a Bi overlayer when the Bi coverage exceeds 1/3ML. At a temperature of 401K, an unusual formation of alloy domains is observed. The origin of this effect is traced down to the mobility of Bi adatoms, which is found to strongly depend on the Bi coverage of the surface alloy on which they reside.