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The ANtarctic Impulsive Transient Antenna (ANITA) searches for ultra high energyneutrinos interacting in the Antarctic ice cap. It is a long duration balloon experimentcomposed of an array of broadband dual-polarized horn antennas that had itsfirst science flight over Antarctica in December 2006 through January 2007. ANITArelies upon the Askaryan effect, in which a particle shower in a dense medium emitscoherent Cherenkov radiation at radio wavelengths, for the detection of a neutrinoinduced shower. Using the null result reported elsewhere of the detection of neutrinocandidates by the ANITA-I experiment, we place a limit upon the flux of neutrinosfrom Gamma Ray Bursts (GRBs). During the ANITA-I flight, 21 GRBs were detectedby spacecraft, but none occurred in the primary ANITA field of view while the instrumentwas operating. Using the measured redshifts of the limited number of localizedGRBs that occurred during the flight, we model their neutrino flux using a basicWaxman-Bahcall model. We are able to place a 90% CL limit on the E-4 promptneutrino fluence from GRB061222B in the energy range of 107.7 GeV ≤ εν ≤ 109.7 GeVof εν4Φν ≤ 4.65x1018 GeV3 cm-2 at a ratio of 4.81x107 to the predicted model for aburst time of 42 seconds. With no neutrino detections over the entire ANITA-I flight,lasting over 17 days, we set a limit with a 90% CL on a E-4 flux over the energyrange of 107.65 GeV ≤ εν ≤ 1010 GeV of εν4Φν ≤ 6.5x1010 GeV3 cm-2 s-1 sr-1.