Data taken during 1997 with the AMANDA-B10 detector are searched for a diffuse flux of neutrinos of all flavors with energies above 10(16) eV. At these energies the Earth is opaque to neutrinos, and thus neutrino induced events are concentrated at the horizon. The background are large muon bundles from down-going atmospheric air shower events. No excess events above the background expectation are observed and a neutrino flux following E-2, with an equal mix of all flavors, is limited to E(2)Phi(10(15) eV < E < 3 x 10(18) eV) less than or equal to 0.99 x 10(-6) GeV cm(-2) s(-1) sr(-1) at 90% confidence level. This is the most restrictive experimental bound placed by any neutrino detector at these energies. Bounds to specific extraterrestrial neutrino flux predictions are also presented.