We report on the first dynamic mechanical studies of blends of conducting polymer poly(aniline) (PANI) in an insulating matrix that exhibit ultralow electrical percolation threshold. Whereas the matrix undergoes an irreversible flow slightly above its glass-rubber transition temperature, blends with PANI mass fraction as low as 1% show a well-defined rubber plateau with a stability range of ca. 100 K. These results corroborate the image of a ramified network of conducting phase, inferred from electrical results. In situ resistance measurements during tensile testing also probe the conductive network. The increase of conductivity in the elastic regime could be interpreted as a consequence of a better alignment of PANI bundles.