This paper contains the complete classification of the finite p-groups G where p is an odd prime, G is generated by two elements, and the commutator subgroup of G is cyclic. These groups are a special kind of two-generator metabelian group, a class that has been studied by Szekeres (1965). He determined the defining relations of such groups but, as he noted, a “residual isomorphism problem” remains. The cyclic commutator groups are simple when considered from this first point of view; they have a short, easily derived set of defining relations. However, the isomorphism problem is a bit complicated for the defining relations contain nine parameters and each of these parameters might and can be an invariant of the group.