Angle-resolved photoemission has been utilized to study the surface electronic structure of 1 / 3 monolayer of Sn on Ge(111) in both the room-temperature (sqrt[3]xsqrt[3] )R30 degrees phase and the low-temperature ( 3x3) charge-density-wave phase. The results reveal a gap opening around the ( 3x3) Brillouin zone boundary, suggesting a Peierls-like transition despite the well-documented lack of Fermi nesting. A highly sensitive electronic response to doping by intrinsic surface defects is the cause for this unusual behavior, and a detailed calculation illustrates the origin of the ( 3x3) symmetry.