  We prove that the KdV-Burgers is globally well-posed in $ H^{-1}(\T) $ with a
solution-map that is analytic from $H^{-1}(\T) $ to $C([0,T];H^{-1}(\T))$
whereas it is ill-posed in $ H^s(\T) $, as soon as $ s<-1 $, in the sense that
the flow-map $u_0\mapsto u(t) $ cannot be continuous from $ H^s(\T) $ to even
${\cal D}'(\T) $ at any fixed $ t>0 $ small enough. In view of the result of
Kappeler and Topalov for KdV it thus appears that even if the dissipation part
of the KdV-Burgers equation allows to lower the $ C^\infty $ critical index
with respect to the KdV equation, it does not permit to improve the $ C^0$
critical index .
