We consider finite groups having a conjugacy class that is the difference of two normal subgroups.  That is, suppose G is a group and M and N are normal subgroups so that N < M, and suppose that there is an element g in G so that the conjugacy class of g is M \ N.  We find a character-theoretic characterization of this condition, and we determine some structural properties of groups with such a conjugacy class.  If we add the condition that M/N is the unique minimal normal subgroup of G/N, then we obtain a generalization of a result by S.M. Gagola.