Methods of controlling round-off error in one-step methods in the numerical solution of ordinary differential equations are compared. A new Algorithm called theoretical cumulative rounding is formulated. Round-off error bounds are obtained for single precision, and theoretical cumulative rounding. Limits of these bounds are obtained as the step length approaches zero. It is shown that the limit of the bound on the round-off error is unbounded for single precision and double precision, is constant for theoretical partial double precision, and is zero for theoretical cumulative rounding. The limits of round-off bounds are not obtainable in actual practice. The round-off error increases for single precision, remains about constant for partial double precision and decreases for cumulative rounding as the step length decreases. Several examples are included. (34 pages)