We present the discovery of > 350 GeV gamma-ray emission from the BL Lacertae ( BL Lac ) object 1ES 2344+514 with the Whipple Observatory 10m gamma-ray telescope . This is the third BL Lac object detected at very high energies ( VHE , E > 300 GeV ) , the other two being Markarian 421 ( Mrk 421 ) and Mrk 501 . These three active galactic nuclei are all X-ray selected and have the lowest known redshifts of any BL Lac objects currently identified . The evidence for emission from 1ES 2344+514 comes mostly from an apparent flare on 1995 December 20 ( universal date ) during which a 6 \sigma excess was detected with an average flux of I ( > 350 GeV ) = 6.6 \pm 1.9 \times 10 ^ { -11 } photons cm ^ { -2 } s ^ { -1 } . This is approximately 63 % of the VHE emission from the Crab Nebula , the standard candle in this field . Observations taken between 1995 October and 1996 January , excluding the night of the flare , yield a 4 \sigma detection indicating a flux level of I ( > 350 GeV ) = 1.1 \pm 0.4 \times 10 ^ { -11 } photons cm ^ { -2 } s ^ { -1 } , or about 11 % of the VHE Crab Nebula flux . Observations taken between 1996 September and 1997 January on this object did not yield a significant detection of a steady flux , nor any evidence of flaring activity . The 99.9 % confidence level upper limit from these observations is I ( > 350 GeV ) < 8.2 \times 10 ^ { -12 } photons cm ^ { -2 } s ^ { -1 } , \lesssim 8 % of the Crab Nebula flux . The low baseline emission level and variation in nightly and yearly flux of 1ES 2344+514 are the same as the VHE emission characteristics of Mrk 421 and Mrk 501 .