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.