The Torelli group of $Wg=\#gSn\times Sn$ is the subgroup of the diffeomorphisms of $Wg$ fixing a disc which act trivially on $Hn(Wg;Z)$. The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $Sp2g(Z)$ or $Og,g(Z)$. In this paper we prove that for $2n\ge 6$ and $g\ge 2$, they are in fact algebraic representations. Combined with previous work, this determines the rational cohomology of the Torelli group in a stable range. We further prove that the classifying space of the Torelli group is nilpotent.