We describe an adaptive Markov chain Monte Carlo method suitable for the estimation of rare failure probabilities in complex probabilistic models. This method, the Accelerated Weight Histogram (AWH) method, has its origin in statistical physics (Lidmar, 2012) and has successfully been applied to molecular dynamics simulations in biophysics. Here we introduce it in the context of structural reliability and demonstrate its usefulness for calculation of failure probabilities in some selected problems of varying degrees of complexity and compare with other established techniques, e.g., subset simulations.