  The recent confirmation that at least some gamma-ray bursters (GRBs) are
indeed at cosmological distances raises the possibility that observations of
these could provide interesting constraints on the fundamental laws of physics.
Here we demonstrate that the fine-scale time structure and hard spectra of GRB
emissions are very sensitive to the possible dispersion of electromagnetic
waves in vacuo with velocity differences $\delta v \sim E/E_{\QG}$, as
suggested in some approaches to quantum gravity. A simple estimate shows that
GRB measurements might be sensitive to a dispersion scale $E_{QG}$ comparable
to the Planck energy scale $E_{P} \sim 10^{19}$ GeV, sufficient to test some of
these theories, and we outline aspects of an observational programme that could
address this goal.
