  We prove that the weakly damped cubic Schr\"odinger flow in $L^2(\T)$
provides a dynamical system that possesses a global attractor. The proof relies
on a sharp study of the behavior of the associated flow-map with respect to the
weak $ L^2(\T) $-convergence inspired by a previous work of the author.
Combining the compactness in $ L^2(\T) $ of the attractor with the approach
developed by Goubet, we show that the attractor is actually a compact set of $
H^2(\T) $. This asymptotic smoothing effect is optimal in view of the
regularity of the steady states.
