  In this article, we present a functional central limit theorem for the paths
of the directed random polymer in space dimension two in the subcritical
intermediate disorder regime. More precisely, the distribution of diffusively
rescaled polymer paths converges in probability to the law of Brownian motion
when taking the weak disorder limit. So far analogous results have only been
established for $d\neq 2$. Along the way, we prove a local limit theorem which
allows us to factorise the point-to-point partition function of the directed
polymer into a product of two point-to-plane partition functions.
