We report the first measurement of the atmospheric electron neutrino flux in the energy range between approximately 80 GeV and 6 TeV, using data recorded during the first year of operation of IceCube's DeepCore low-energy extension. Techniques to identify neutrinos interacting within the DeepCore volume and veto muons originating outside the detector are demonstrated. A sample of 1029 events is observed in 281 days of data, of which 496 +/- 66(stat) +/- 88(syst) are estimated to be cascade events, including both electron neutrino and neutral current events. The rest of the sample includes residual backgrounds due to atmospheric muons and charged current interactions of atmospheric muon neutrinos. The flux of the atmospheric electron neutrinos is consistent with models of atmospheric neutrinos in this energy range. This constitutes the first observation of electron neutrinos and neutral current interactions in a very large volume neutrino telescope optimized for the TeV energy range. DOI: 10.1103/PhysRevLett.110.151105