The mathematical assumption of complete frequency redistribution (CFR) is investigated for the problem of the transport of resonance radiation through gaseous media of large optical thicknesses. The investigation is initiated by solving the CFR integral equation of radiative transfer in a plane-parallel layer of gas. A steady-state excited-atom density is maintained by plane-wave white-light radiation produced by a source external to the medium. Computer solutions of the CFR transfer equation are obtained for various values of the natural damping coefficient and a range of total optical thicknesses. The CFR source functions for large total optical thicknesses are then used as trial solutions in the exact integral equation for the problem. Only a first iteration is performed. The resultant intensity profiles for various values of the damping coefficient are compared with CFR profiles calculated from the CFR source functions. The orders of magnitude of the total optical thicknesses and damping coefficients used in the numerical computations approximate the values that are realized in planetary nebulae. The non-CFR intensity profiles calculated by the method described above deviate from the CFR profiles in the manner predicted by qualitative argument.