  The photo-meson production of pions by shock-accelerated protons could
generate a burst of ~10^14 eV neutrinos from gamma-ray bursts (GRBs) observable
in the Antarctic Muon and Neutrino Detector Array (AMANDA) or its larger
successors. Measurement of this flux can test the hypothesis that GRBs are the
sources of the highest-energy cosmic rays, and GRB neutrinos could permit
high-precision experiments in neutrino limiting speed, neutrino oscillations,
and the weak equivalence principle. Neutrino emission can be expected primarily
during the prompt gamma-ray flash and satellite coincidence provides a
well-defined window in position and time that can be searched for an excess of
upgoing muon events in AMANDA from bursts in the Northern Hemisphere. Using an
event quality analysis to further reduce background in a sample of 78 GRBs from
the 1997 AMANDA-B10 data set, I find a fluence limit of (E_nu)^2
(dN_nu)/(dE_nu)<3.8*10^-4 min(1,E_nu/E_break) [TeV cm^-2] per average burst,
which is orders of magnitude more stringent than in similar previous searches.
