An exact solution is obtained for the energy-dependent Boltzmann transport equation for thermal neutrons near a temperature discontinuity. The medium is nonabsorbing and infinite. The method of solution consists of expanding the energy-transfer kernel in a degenerate form and then solving directly for the solutions of the resulting homogeneous equation. Both discrete and singular solutions are found. The angular flux is then expanded in terms of a complete set of these solutions. Finally, the expansion coefficients are determined by applying the boundary conditions associated with the temperature-discontinuity problem. Numerical calculations of both scalar neutron flux and total neutron density are included for various temperature ratios and neutron-to-moderator mass ratios. A comparison of the transport-theory results with diffusion-theory results shows that diffusion theory describes the neutron flux accurately for small values of temperature discontinuity. Diffusion-theory calculations become less accurate, however, as the higher energy modes become important. 81 pp. Ref.