We continue with the study of the mollified stochastic heat equation in $d\geq 3$ given by $d u_{ε,t}=\frac 12Δu_{ε,t}+ βε^{(d-2)/2} \,u_{ε,t} \,d B_{ε,t}$ 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 $β>0$ in the limiting object of the smoothened solution $u_ε$ as the smoothing parameter $ε\to 0$ This partition function naturally defines a quenched polymer path measure and we prove that as long as $β>0$ stays small enough while $u_ε$ 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.