  In this paper we consider the long time behavior of the weakly damped, forced
Korteweg-de Vries equation in the Sololev spaces of the negative indices in the
periodic case. We prove that the solutions are uniformly bounded in
$\dot{H}^s(\T)$ for $s>-\dfrac{1}{2}$. Moreover, we show that the solution-map
possesses a global attractor in $\dot{H}^s(\T)$ for $s>-\dfrac{1}{2}$, which is
a compact set in $H^{s+3}(\T)$.
