Data from the AMANDA-B10 detector taken during the austral winter of 1997 have been searched for a diffuse flux of high energy extraterrestrial muon neutrinos. This search yielded no excess events above those expected from background atmospheric neutrinos, leading to upper limits on the extraterrestrial neutrino flux measured at the earth. For an assumed [Formula presented] spectrum, a [Formula presented] classical confidence level upper limit has been placed at a level [Formula presented] (for a predominant neutrino energy range 6–1000 TeV), which is the most restrictive bound placed by any neutrino detector. Some specific predicted model spectra are excluded. Interpreting these limits in terms of the flux from a cosmological distributions of sources requires the incorporation of neutrino oscillations, typically weakening the limits by a factor of 2.