The variation in space of the thermal-neutron spectrum close to a temperature discontinuity in a non-absorbing free gas has been calculated in the diffusion approximation on a G-20 computer. This spectrum forms a basis for testing two simple rethermalization models. The two-overlapping-group model due to Selengut gives very good agreement with the computer calculations when used with an improved transfer cross section. The single-Maxwellian model in which the neutron temperature varies exponentially with distance from the discontinuity does not give as good agreement.