The author discusses a new method for interpolation between mesh points of Runge-Kutta algorithms for the approximate solution of ordinary differential equations. The method is shown to fall under the classification of scaled Runge-Kutta algorithms as considered by \textit{M. K. Horn} [ibid. 20, 558-568 (1983; Zbl 0511.65048)]. The interpolation procedure utilizes certain computed values from the Runge-Kutta routine and Hermite interpolation from these values. It is proved that the local order of accuracy agrees with the order of accuracy of the corresponding Runge-Kutta algorithm and that the output is globally smooth. The method is described explicitly for a particular four stage fourth order algorithm given by \textit{R. England} [Computer J. 12, 166-170 (1969; Zbl 0182.219)].