A new relaxation mechanism is shown to arise from overdamped two-level systems above a critical temperature T * ≈ 5 K, thus yielding an explanation for experimental observations in dielectric glasses in the temperature range between T * and the relaxation peak at 50 K. Using the distribution function of the tunnelling model for the parameters of the two-level systems, both the linear decrease of the sound velocity and the linear increase of the absorption up to the relaxation maximum, are quantitatively accounted for by our theory.PACS. 61.40-amorphous and polymeric materials PACS. 63.50 -disordered solids, vibrational states PACS. 77.22G -relaxation phenomena, dielectrics