We prove that the truncated correlation functions of the charge and gradient fields associated with the massless sine-Gordon model on $\mathbb{R}^2$ with $β=4π$ exist for all coupling constants and are equal to those of the chiral densities and vector current of free massive Dirac fermions. This is an instance of Coleman's prediction that the massless sine-Gordon model and the massive Thirring model are equivalent (in the above sense of correlation functions). Our main novelty is that we prove this correspondence starting from the Euclidean path integral in the non-perturative regime of the infinite volume models. We use this correspondence to show that the correlation functions of the massless sine-Gordon model with $β=4π$ decay exponentially and that the corresponding probabilistic field is localized.