A new predictor for use with the Adams-Moulton corrector has been devet.oped.Truncation errors at each step arc determined, to first order, solely by the characteristics of the corrector.Likewise, the propagation of error in the evaluation of definite integrals is dependent only on the corrector equation.(The only purpose of the predictor here is to form an error estiinate.)The predictor equation and the corrector equation are independently and jointly of the fourth order.The predictor equation developed here is believed to have the largest range of absolute stability (including h = 0) for the combined predictor-corrector algorithm that is possible.At the same time the method has a range of relative stability which will maintaiil stable propagation of relative errors when truncation errors of less than one part in one thousand are being incurred.The storage required for previous derivative values is no greater than that for the standard Adams-Moulton method with the Adams-Bashforth predictor.