The asymptotic energy spectrum of thermal neutrons in an infinite medium of beryllium has been calculated for three temperatures: 300°K, 200°K, 100°K, for a constant plane source of neutrons at the midplane. The techniques of multigroup diffusion theory were applied, using Nelkin's first order scattering kernel for Be, and the energy-dependent transport mean free path, λtr(E), calculated by Bhandari. Because of the violent variation of λtr in the vicinity of the Bragg cutoff energy, for the lower moderator temperatures the calculated flux spectrum is quite different from the Maxwellian. At 300°K the deviation from the Maxwellian is small.