We have found an exact expression for the decoherence rate of a qubit coupled to fluctuating background charges. At low temperatures T the decoherence rate is linear in T while at high temperatures our expression coincides with a known classical solution which is, however, reached in the strong coupling regime at surprisingly high T. A comparison of experimental data with our predictions would give an unambiguous indication of whether the experimentally observed decoherence of a Josephson-junction qubit is due to background charges.