  We consider a system of $N$-Bosons with a two-body interaction potential $V
\in L^2(\mathbb{R}^3)+L^\infty(\mathbb{R}^3)$, possibly singular than the
Coulomb interaction. We show that, with $H^1(\mathbb{R}^3)$ initial data, the
difference between the many-body Schr\"odinger evolution in the mean-field
regime and the corresponding Hartree dynamics is of order $1/N$, for any fixed
time. The $N$-dependence of the bound is optimal.
