The inverse square law of gravity is poorly probed by experimental tests at distances of ~10 AU. Recent analysis of the trajectory of the Pioneer 10 and 11 spacecraft have shown an unmodeled acceleration directed toward the Sun, which was not explained by any obvious spacecraft systematics, and occurred when at distances greater than 20 AU from the Sun. If this acceleration represents a departure from Newtonian gravity or is indicative of an additional mass distribution in the outer solar system, it should be detectable in the orbits of trans-Neptunian objects (TNOs). To place limits on deviations from Newtonian gravity, we have selected a well-observed sample of TNOs found orbiting between 20 and 100 AU from the Sun. By examining their orbits with modified orbital fitting software, we place tight limits on the perturbations of gravity that could exist in this region of the solar system.