  For a quantum spin chain or 1D fermionic system, we prove that the Drude
weight D verifies the universal Luttinger liquid relation $v_s^2=D/\kappa$,
where $\kappa$ is the susceptibility and $v_s$ is the Fermi velocity. This
result is proved by rigorous Renormalization Group methods and is true for any
weakly interacting system, regardless its integrablity. This paper, combined
with a previous paper of the same authors, completes the proof of the Luttinger
liquid conjecture for such systems.
