  We present constraints derived from a search of four years of IceCube data
for a prompt neutrino flux from gamma-ray bursts (GRBs). A single
low-significance neutrino, compatible with the atmospheric neutrino background,
was found in coincidence with one of the 506 observed bursts. Although GRBs
have been proposed as candidate sources for ultra-high energy cosmic rays, our
limits on the neutrino flux disfavor much of the parameter space for the latest
models. We also find that no more than $\sim1\%$ of the recently observed
astrophysical neutrino flux consists of prompt emission from GRBs that are
potentially observable by existing satellites.
