The localization phenomena exhibited by a commonly studied model of electrons in a substitutional alloy are found to be much richer than has previously been reported. A nondegenerate tight-binding band arising primarily from one constituent of the alloy is considered. It may possess states localized in small isolated clusters (as studied in percolation theory), and localized states decaying exponentially at large distances (as in the Anderson model), plus a third class of localized states which have not been discussed before. These last states occur at special energies, but are not isolated from the bulk of the material. Examples are constructed, and an approximate calculation of their spectral weight is carried out. This enumeration provides an explanation of the anomalies which have appeared in recent numerical calculations of the density of states of this model. The continuous part of the spectrum also displays surprising behavior, as revealed by numerical calculations. There appears to be a "forbidden energy" at the center of the band, at which the density of states goes continuously to zero. The eigenstates at nearby energies appear to be localized in the Anderson sense. The model exhibits not one, but several pairs of mobility edges, separating localized from extended states. These calculations make available nontrivial exact results for a three-dimensional disordered material, as a test of approximate theories. None of the existing theories predict these effects. In addition, we suggest several real systems in which the new class of localized states, and perhaps the associated effects on the continuous spectrum, can be observed. © 1972 The American Physical Society.