We have measured the rate of thermally assisted magnetization reversal of submicron-sized magnetic thin films. For fields H just less than the zero-temperature switching field H(C), the probability of reversal, P(exp)(s)(t), increases for short times t, achieves a maximum value, and then decreases exponentially. Micromagnetic simulations exhibit the same behavior and show that the reversal proceeds through the annihilation of two domain walls that move from opposite sides of the sample. The behavior of P(exp)(s)(t) can be understood through a simple "energy-ladder" model of thermal activation.