<p>The need for high-order accurate and efficient numerical methods cannot be overemphasized. This article proposes such a method for initial value problems of ordinary differential equations by suggesting a fourth-order accurate algorithm with detailed theoretical analysis and numerical verification. First, the differential problem is converted to an integral equation. Then, numerical quadrature rule is used to transform the result to a fully discrete problem. The implicitness of the discrete problem necessitates the formulation of an explicit predictor which results to a four-step predictor-corrector method. Truncation error analysis is used to prove consistency; stability is also established with respect to perturbation in the initial data. Then, a new discrete Gronwall inequality is proposed, and used, to present a rigorous convergence analysis, establishing the fourth-order accuracy of the method. Seven numerical experiments are conducted and used to demonstrate that the method (i) is fourth-order accurate as theoretically proved, (ii) is very much more computationally efficient than the Runge-Kutta method, and (iii) is more competitive, in terms of accuracy, than the Hamming method. Therefore, the method achieves the desired objective of being very high-order accurate and efficient at the same time.</p>