This paper develops a numerical technique to price barrier options for which monitoring of the barrier condition is done at discrete time intervals. The discrete-time multidimensional integration required to implement neutral-risk pricing of the securities is accomplished by sequential numerical integration. Approximate prices are obtained by one-dimensional quadrature combined with polynomial approximation, avoiding recursive application of numerical integration and the attendant exponential increase in computational effort. The number of calculations required is linear in the number of monitoring dates, affording significant computational economy. The method is demonstrated by application to a variety of barrier options. It compares favorably with existing methods in both computation time and accuracy.