  We constrain the possibility of a non-trivial refractive index in free space
corresponding to an energy-dependent velocity of light: c(E) \simeq c_0 (1 -
E/M), where M is a mass scale that might represent effect of
quantum-gravitational space-time foam, using the arrival times of sharp
features observed in the intensities of radiation with different energies from
a large sample of gamma-ray bursters (GRBs) with known redshifts. We use
wavelet techniques to identify genuine features, which we confirm in
simulations with artificial added noise. Using the weighted averages of the
time-lags calculated using correlated features in all the GRB light curves, we
find a systematic tendency for more energetic photons to arrive earlier.
However, there is a very strong correlation between the parameters
characterizing an intrinsic time-lag at the source and a distance-dependent
propagation effect. Moreover, the significance of the earlier arrival times is
less evident for a subsample of more robust spectral structures. Allowing for
intrinsic stochastic time-lags in these features, we establish a statistically
robust lower limit: M > 0.9x10^{16} GeV on the scale of violation of Lorentz
invariance.
