Interfacial flows close to a moving contact line are inherently multiscale. The shape of the interface and the flow at meso- and macroscopic scales inherit an apparent interface slope and a regularization length, both named after Voinov, from the microscopic inner region. Here, we solve the inner problem associated with the contact line motion for a volatile fluid at equilibrium with its vapor. The evaporation or condensation flux is then controlled by the dependence of the saturation temperature on interface curvature—the so-called Kelvin effect. We derive the dependencies of the Voinov angle and of the Voinov length as functions of the parameters of the problem. We then identify the conditions under which the Kelvin effect is indeed the mechanism regularizing the contact line motion.