We describe a method by which gravitational wave observations of eccentric binary systems could be used to test general relativity's prediction that gravitational waves are dispersionless. We present our results in terms of the graviton having a nonzero rest mass, or equivalently a noninfinite Compton wavelength. We make a rough estimate of the bounds that might be obtained following gravitational wave detections by the Laser Interferometer Space Antenna. The bounds we find are comparable to those obtainable from a method proposed by C. M. Will and several orders of magnitude stronger than other dynamic (i.e., gravitational wave-based) tests that have been proposed. The method described here has the advantage over those proposed previously of being simple to apply, as it does not require the in-spiral to be in the strong field regime nor correlation with electromagnetic signals. We compare our results with those obtained from static (i.e., non-gravitational wave-based) tests
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