The time taken for positrons from a radioactive source to reach a velocity distribution in the near-thermal range has been measured for mixtures of nitrogen gas with argon, and of methane with argon. An analysis of the shoulder breadths for the free positron components leads to slowing down times of 16 ns for nitrogen and 0.25 ns for methane at densities of 1 amagat. The long slowing time for nitrogen must occur during the last eV or so of slowing and the dominant mode of energy loss is attributable to rotational excitation of the molecules. For methane the slowing to the threshold for vibrational excitation must be extremely rapid, but there is no clearly established mechanism which could account for the remaining energy loss taking place so rapidly.