A modified phase-fitted Runge–Kutta method (i.e., a method with phase-lag of order infinity) for the numerical solution of periodic initial-value problems is constructed in this paper. This new modified method is based on the Runge–Kutta fifth algebraic order method of Dormand and Prince [33]. The numerical results indicate that this new method is more efficient for the numerical solution of periodic initial-value problems than the well known Runge–Kutta method of Dormand and Prince [33] with algebraic order five.