  We consider the KPZ equation in space dimension 2 driven by space-time white
noise. We showed in previous work that if the noise is mollified in space on
scale $\epsilon$ and its strength is scaled as $\hat\beta / \sqrt{|\log
\epsilon|}$, then a transition occurs with explicit critical point $\hat\beta_c
= \sqrt{2\pi}$. Recently Chatterjee and Dunlap showed that the solution admits
subsequential scaling limits as $\epsilon \downarrow 0$, for sufficiently small
$\hat\beta$. We prove here that the limit exists in the entire subcritical
regime $\hat\beta \in (0, \hat\beta_c)$ and we identify it as the solution of
an additive Stochastic Heat Equation, establishing so-called Edwards-Wilkinson
fluctuations. The same result holds for the directed polymer model in random
environment in space dimension 2.
