In this paper, a Monte Carlo method for the calculation of the change in the neutron multiplication factor of a reactor due to cross-section perturbations is developed. Although similar to the perturbation source method developed by Matthes, this method is not limited to problems in which first-order perturbation theory is applicable. This method has been implemented in the KENO computer code and applied to a variety of problems. The results of these calculations are presented in this paper. This approach should prove useful in the solution of problems in which other Monte Carlo methods, such as Matthes' first-order perturbation source method and correlated sampling, fail.