The Kramers theory for the thermally activated rate of escape of a Brownian particle from a potential well is extended to a barrier of arbitrary shape. The extension is based on an approximate solution of the underlying Fokker-Planck equation in the spatial diffusion regime. With the use of the Mel＇nikov-Meshkov result for the underdamped Brownian motion an overall rate expression is constructed, which interpolates the correct limiting behavior for both weak and strong friction. It generalizes in a natural way various different rate expressions that are already available in the literature for parabolic, cusped, and quartic barriers. Applications to symmetric parabolic and cusped double-well potentials show good agreement between the theory and estimates of the rates from numerical calculations.