An effective multistep algorithm for numerical solution of Volterra integral equations of the second kind, based on the implicit Runge-Kutta (RK) method, is constructed. The choice of the Gauss scheme for the implicit RK method allows to obtain algorithm, having a higher approximation order for one-step method and maintaining the same accuracy for most calculations in multistep method. The comparison of solutions computed by algorithm, presented in this paper, with the analogues ones, obtaining earlier by explicit RK method, reveales more precise results for the first of them and possibility to integrate with much more longer steps without essential loss of accuracy.