This paper deals with adapting Runge-Kutta methods to differential equations with a lagging argument. A new interpolation procedure is introduced which leads to numerical processes that satisfy an important asymptotic stability condition related to the class of testproblemsU′(t)=λU(t)+μU(t−τ) with λ, μ ε C, Re(λ)<−|μ|, and τ>0. Ifc _i denotes theith abscissa of a given Runge-Kutta method, then in thenth stept _n−1→t _n :=t _n−1+h of the numerical process our interpolation procedure computes an approximation toU(t _n−1+c _i h−τ) from approximations that have already been generated by the process at pointst _j−1+c _i h(j=1,2,3,...). For two of these new processes and a standard process we shall consider the convergence behaviour in an actual application to a given, stiff problem.