In a previous paper the authors suggested that the accurate correctors proposed by Gragg and Stetter for solving ordinary differential equations should be accompanied by similar predictors. In each method in that paper the corrector and one of the predictors use one “nonstop” point within the interval of integration. In the present paper a corrector is dealt with in which two “nonstep” points are used, and in which, to some degree, the authors have “balanced” the contributions of the errors in the predictors to the total local truncation error, a technique due to Butcher.