The BL Lacertae (BL Lac) object 1ES 2344+514 (1ES 2344), at a redshift of 0.044, was discovered as a source of very high energy (VHE) gamma rays by the Whipple collaboration in 1995 (Catanese et al.). This detection was recently confirmed by the HEGRA collaboration (Tluczykont et al.). As is typical for high-frequency-peaked blazars, the VHE gamma-ray emission is highly variable. On the night of 1995 December 20, a gamma-ray flare of 5.3 σ significance was detected, the brightest outburst from this object to date. The emission region is compatible with a point source. The spectrum between 0.8 and 12.6 TeV can be described by a power law, If we compare the spectral index with that of the other five confirmed TeV blazars, the spectrum of 1ES 2344 is similar to that of 1ES 1959+650, which is located at almost the same distance. The spectrum of 1ES 2344 is steeper than the brightest flare spectra of Markarian 421 (Mrk 421) and Markarian 501 (Mrk 501), both of which are located at a distance about two-thirds that of 1ES 2344, and harder than the spectra of PKS 2155-304 and H1426+428, which are located almost 3 times as far away. This trend is consistent with attenuation caused by the infrared extragalactic background radiation.