Since the discovery of the cosmological origin of gamma-ray bursts (GRBs), there has been growing interest in using these transient events to probe the quantum gravity energy scale in the range 1016-1019 GeV, up to the Planck mass scale. This energy scale can manifest itself through a measurable modification in the electromagnetic radiation dispersion relation for high-energy photons originating from cosmological distances. We have used data from the GRB of 2002 December 6 (GRB 021206) to place an upper bound on the energy dispersion of the speed of light. The limit on the first-order quantum gravity effects derived from this single GRB indicates that the energy scale is in excess of 1.8×1017 GeV. We discuss a program to further constrain the energy scale by systematically studying such GRBs.