The thermally activated escape of a Brownian particle over an arbitrarily shaped potential barrier is considered. Based on an approximate solution of the corresponding Fokker-Planck equation a rate expression is given. It agrees in the limiting case of high friction with the rate following from the corresponding Smoluchowski equation and, in the limit of weak friction with the rate obtained from transition state theory. For a parabolic barrier the approximate rate expression deviates less than 16% from the known result. The results for cusp shaped and quartic barriers agree with known expressions which have been obtained by other means. Estimates of the rates from numerical simulations are compared with the approximate rate expressions for the cusp and quartic barrier.