In this paper we present a method for solution of the equation of transfer for elliptically polarized radiation in an illuminated stellar atmosphere. Although the method is completely general, it is specifically applied to the gray-atmosphere problem in order to demonstrate its accuracy and stability. The main advantage of the method is the separation of the depth dependence of the solution from the angular dependence in situations containing little, if any, symmetry. It is shown that the method can reproduce solutions for problems in which independent solutions are available. Furthermore, the method will handle problems in which the illuminating radiation comes from a finite solid angle over which the defining properties of the radiation field are allowed to vary.