We consider a drift-diffusion process on a smooth potential landscape with small noise. We give a new proof of the Eyring-Kramers formula which asymptotically characterizes the spectral gap of the generator of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg and of the mean-difference estimate introduced by Chafai and Malrieu. The new proof exploits the idea that the process has two natural time-scales: a fast time-scale resulting from the fast convergence to a metastable state, and a slow time-scale resulting from exponentially long waiting times of jumps between metastable states. A nice feature of the argument is that it can be used to deduce an asymptotic formula for the log-Sobolev constant, which was previously unknown.