Aiming to the weakly discontinuous optimal control problems, an adaptive pseudospectral method is proposed. Some point sequences divide the time interval into several subintervals. The control and state functions are considered as piecewise continuous polynomials. On the foundation of the convergence of numerical solutions, it is proved that the point sequences dividing the time interval can converge to the weakly discontinuous points. According to the Cauchy convergence principle, the positions of the weakly discontinuous points can be estimated by the changes of the foregoing point sequences, and therefore an error indicator is designed to adjust the subintervals and the degrees of the approximating polynomials. Several examples are given to compare the proposed method with two pseudospectral methods, and the results show that the accuracy and effectiveness of the proposed method outperform those of the other two methods.