When the perturbing planets are Uranus and Neptune, the perturbations on comets are so much weaker than with Jupiter and Saturn that a study of the comets’ orbital evolution, using exact numerical integration, would require 200 times more revolutions. This is hardly practical with present computers. Here we describe results with a simulation approach, the “Monte Carlo (random walk) method.” The proper distribution shape for the perturbations in energy are found from a few thousand numerical integrations, then this distribution of perturbations is applied to millions of simulated orbit-revolutions. This method reproduces earlier Jupiter results in 1/500 the former computation time. We find that Neptune can capture near-parabolic comets with perihelia in the range of 30 to 34 AU, increasing their 1/a-values and decreasing their perihelia until they reach a region where Uranus can interact. Uranus in turn passes some of these on to Saturn, who passes some to Jupiter. Ultimately a few reach the orbits of the visible short-period comets. The process requires about 200,000 comet orbit-revolutions, 4 × 108 years, and the efficiency is one in 6000. The rest of the comets are ejected on hyperbolic orbits.