The Antarctic Muon and Neutrino Detector Array (AMANDA) began collecting data with ten strings in 1997. Results from the first year of operation are presented. Neutrinos coming through the Earth from the Northern Hemisphere are identified by secondary muons moving upward through the array. Cosmic rays in the atmosphere generate a background of downward moving muons, which are about 10^6 times more abundant than the upward moving muons. Over 130 days of exposure, we observed a total of about 300 neutrino events. In the same period, a background of 1.05*10^9 cosmic ray muon events was recorded. The observed neutrino flux is consistent with atmospheric neutrino predictions. Monte Carlo simulations indicate that 90% of these events lie in the energy range 66 GeV to 3.4 TeV. The observation of atmospheric neutrinos consistent with expectations establishes AMANDA-B10 as a working neutrino telescope.