The low temperature electrical conductivities of heavily iodine-doped polyacetylene with different degrees of disorder are reported. The sample with room-temperature conductivity of 1.0 x 10(5) S/cm showed weak temperature dependence below 1 K down to 0.8 mK. With moderate increase of disorder, logarithmic temperature dependence was observed. The results are interpreted in terms of scattering of conduction carriers accompanying change of atomic configurations of equivalent energy in the polymer backbone, after similar process in amorphous metal alloys.