We consider the dynamics of an axisymmetric, partially wetting droplet of a one-component volatile liquid. The droplet is supported on a smooth superheated substrate and evaporates into a pure vapour atmosphere. In this process, we take the liquid properties to be constant and assume that the vapour phase has poor thermal conductivity and small dynamic viscosity so that we may decouple its dynamics from the dynamics of the liquid phase. This leads to a so-called ‘one-sided’ lubrication-type model for the evolution of the droplet thickness, which accounts for the effects of evaporation, capillarity, gravity, slip and kinetic resistance to evaporation. By asymptotically matching the flow near the contact line region and the bulk of the droplet in the limit of a small slip length and commensurably small evaporation and kinetic resistance effects, we obtain coupled evolution equations for the droplet radius and volume. The predictions of our asymptotic analysis, which also include an estimate of the evaporation time, are found to be in excellent agreement with numerical simulations of the governing lubrication model for a broad range of parameter regimes.