The hazard rate function of the Weibull Exponential lifetime distribution is decreasing over time up to a change point and remains a constant afterward We derive some of the properties of this lifetime distribution and develop a Bayes sequential procedure for estimating the change point. The problem of determining an appropriate burn in time under a certian loss function is also considered. An adaptive Bayesian decision rule for the termination of the burn in procedure is proposed. Results of numerical simulations are given to demonstrate the proposed procedure.