Using the neutrino telescope AMANDA-II, we have conducted two analyses searching for neutrino-induced cascades from gamma-ray bursts. No evidence of astrophysical neutrinos was found, and limits are presented for several models. We also present neutrino effective areas which allow the calculation of limits for any neutrino production model. The first analysis looked for a statistical excess of events within a sliding window of 1 or 100 seconds (for short and long burst classes, respectively) during the years 2001-2003. The resulting upper limit on the diffuse flux normalization times E^2 for the Waxman-Bahcall model at 1 PeV is 1.6 x 10^-6 GeV cm^-2 s^-1 sr^-1 (a factor of 120 above the theoretical prediction). For this search 90% of the neutrinos would fall in the energy range 50 TeV to 7 PeV. The second analysis looked for neutrino-induced cascades in coincidence with 73 bursts detected by BATSE in the year 2000. The resulting upper limit on the diffuse flux normalization times E^2, also at 1 PeV, is 1.5 x 10^-6 GeV cm^-2 s^-1 sr^-1 (a factor of 110 above the theoretical prediction) for the same energy range. The neutrino-induced cascade channel is complementary to the up-going muon channel. We comment on its advantages for searches of neutrinos from GRBs and its future use with IceCube
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