  We continue with the study of the mollified stochastic heat equation in
$d\geq 3$ given by $d u_{\epsilon,t}=\frac 12\Delta u_{\epsilon,t}+ \beta
\epsilon^{(d-2)/2} \,u_{\epsilon,t} \,d B_{\epsilon,t}$ with spatially
smoothened cylindrical Wiener process $B$, whose (renormalized) Feynman-Kac
solution describes the partition function of the continuous directed polymer.
In an earlier work (\cite{MSZ16}), a phase transition was obtained, depending
on the value of $\beta>0$ in the limiting object of the smoothened solution
$u_\epsilon$ as the smoothing parameter $\epsilon\to 0$ This partition function
naturally defines a quenched polymer path measure and we prove that as long as
$\beta>0$ stays small enough while $u_\epsilon$ converges to a strictly
positive non-degenerate random variable, the distribution of the diffusively
rescaled Brownian path converges under the aforementioned polymer path measure
to standard Gaussian distribution.
