We describe an adaptive dose escalation scheme for use in cancer phase I clinical trials. The method is fully adaptive, makes use of all the information available at the time of each dose assignment, and directly addresses the ethical need to control the probability of overdosing. It is designed to approach the maximum tolerated dose as fast as possible subject to the constraint that the predicted proportion of patients who receive an overdose does not exceed a specified value. We conducted simulations to compare the proposed method with four up-and-down designs, two stochastic approximation methods, and with a variant of the continual reassessment method. The results showed the proposed method effective as a means to control the frequency of overdosing. Relative to the continual reassessment method, our scheme overdosed a smaller proportion of patients, exhibited fewer toxicities and estimated the maximum tolerated dose with comparable accuracy. When compared to the non-parametric schemes, our method treated fewer patients at either subtherapeutic or severely toxic dose levels, treated more patients at optimal dose levels and estimated the maximum tolerated dose with smaller average bias and mean squared error. Hence, the proposed method is promising alternative to currently used cancer phase I clinical trial designs.