  We prove an optimal order error bound in the discrete $H^2(\Omega)$ norm for
finite difference approximations of the first boundary-value problem for the
biharmonic equation in $n$ space dimensions, with $n \in \{2,\dots,7\}$, whose
generalized solution belongs to the Sobolev space $H^s(\Omega) \cap
H^2_0(\Omega)$, for $\frac{1}{2} \max(5,n) < s \leq 4$, where $\Omega =
(0,1)^n$. The result extends the range of the Sobolev index $s$ in the best
convergence results currently available in the literature to the maximal range
admitted by the Sobolev embedding of $H^s(\Omega)$ into $C(\overline\Omega)$ in
$n$ space dimensions.
