We present the discovery of gamma-ray emission greater than 350 GeV from the BL Lacertae (BL Lac) object 1ES 2344+514 with the Whipple Observatory 10 m 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 with declination greater than 0°. The evidence for emission from 1ES 2344+514 comes mostly from an apparent flare on 1995 December 20 (UT) during which a 6 σ excess was detected with an average flux of I(>350 GeV) = 6.6 ± 1.9 × 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 σ detection indicating a flux level of I(>350 GeV) = 1.1 ± 0.4 × 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 or any evidence of flaring activity. The 99.9% confidence level upper limit from these observations is I(>350 GeV) < 8.2 × 10-12 photons cm-2 s-1, lesssim8% of the Crab Nebula flux. The low baseline emission level and variation in the nightly and yearly flux of 1ES 2344+514 are the same as the VHE emission characteristics of Mrk 421 and Mrk 501.