  In this article, we present an invariance principle 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.
