  We calculate the average conductivity sigma (omega) of interacting electrons
in one dimension in the presence of a long-range random potential (forward
scattering disorder). Taking the curvature of the energy dispersion into
account, we show that weak disorder leads to a transport scattering rate that
vanishes as omega^2 for small frequency omega. This implies that the real part
of the conductivity remains finite for omega -> 0, while the imaginary part
diverges. These effects are lost within the usual bosonization approach, which
relies on the linearization of the energy dispersion. We discuss our result in
the light of a recent experiment.
