This paper describes the search for astronomical sources of high-energy neutrinos using the AMANDA-B10 detector, an array of 302 photomultiplier tubes used for the detection of Cerenkov light from upward-traveling neutrino-induced muons, buried deep in ice at the South Pole. The absolute pointing accuracy and angular resolution were studied by using coincident events between the AMANDA detector and two independent telescopes on the surface, the GASP air Cerenkov telescope and the SPASE extensive air shower array. Using data collected from 1997 April to October (130.1 days of live time), a general survey of the northern hemisphere revealed no statistically significant excess of events from any direction. The sensitivity for a flux of muon neutrinos is based on the effective detection area for through-going muons. Averaged over the northern sky, the effective detection area exceeds 10,000 m2 for E μ ≈ 10 TeV. Neutrinos generated in the atmosphere by cosmic-ray interactions were used to verify the predicted performance of the detector. For a source with a differential energy spectrum proportional to Eν-2 and declination larger than +40°, we obtain E2(dNν/dE) ≤ 10-6 GeV cm-2 s-1 for an energy threshold of 10 GeV.