  We prove the existence of the global attractor in $ \dot H^s$, $s > 11/12$
for the weakly damped and forced mKdV on the one dimensional torus. The
existence of global attractor below the energy space has not been known, though
the global well-posedness below the energy space is established. We directly
apply the I-method to the damped and forced mKdV, because the Miura
transformation does not work for the mKdV with damping and forcing terms. We
need to make a close investigation into the trilinear estimates involving
resonant frequencies, which are different from the bilinear estimates
corresponding to the KdV.
