We report results of low-temperature thermodynamic and transport measurements of Pb1-xTlxTe single crystals for Tl concentrations up to the solubility limit of approximately x=1.5%. For all doped samples, we observe a low-temperature resistivity upturn that scales in magnitude with the Tl concentration. The temperature and field dependence of this upturn are consistent with a charge Kondo effect involving degenerate Tl valence states differing by two electrons, with a characteristic Kondo temperature T(K) approximately 6 K. The observation of such an effect supports an electronic pairing mechanism for superconductivity in this material and may account for the anomalously high T(c) values.