In this paper we investigate extended inflation with an exponential potential $V(\sigma)= V_0 e^{-\kappa\sigma}$, which provides a simple cosmological scenario where the distribution of the constants of Nature is mostly determined by $\kappa$. In particular, we show that this theory predicts a uniform distribution for the Planck mass at the end of inflation, for the entire ensemble of universes that undergo stochastic inflation. Eternal inflation takes place in this scenario for a broad family of initial conditions, all of which lead up to the same value of the Planck mass at the end of inflation. The predicted value of the Planck mass is consistent with the observed value within a comfortable range of values of the parameters involved.