Our earlier treatment of the energy-dependent transport equation is extended to include the case in which cross sections are functions of energy. The technique again consists of finding solutions to the homogeneous transport equation after expansion in terms of a complete set of functions in the energy variable. Unlike the problem treated earlier, the full-range completeness theorem for these eigenfunctions requires the solution of a coupled set of singular integral equations. This solution is effected by a generalization of a trick used by Case and is applied to the problem for the infinite-medium Green's function. Numerical results are given for a heavy gas model. The half-range completeness theorem, which may be applied to half-space and finite slab problems, is proven in a companion paper.