Cerium adatoms, deposited on a Ag(111) surface, are found by low-temperature scanning tunneling microscopy to self-assemble into large ordered hexagonal arrays covering macroscopically the entire surface. We show that the 32 A periodicity of the superlattice is caused by the interaction of surface-state electrons with Ce adatoms and that the large-scale formation of the superlattice is governed by a subtle balance between the sample temperature, the surface diffusion barrier, and the concentration-dependent adatom interaction potential.