A quantitative study of the Luminous Blue Variable NGC 2363-V1 in the Magellanic galaxy NGC 2366 (D = 3.44 Mpc) is presented, based on ultraviolet and optical HST/STIS spectroscopy. Contemporary WFPC2 and WHT imaging reveals a modest V-band brightness increase of ~ 0.2 mag per year between 1996 January and 1997 November, reaching V=17.4 mag, corresponding to Mv=-10.4 mag. Subsequently, V1 underwent a similar decrease in V-band brightness, together with a UV brightening of 0.35 mag from 1997 November to 1999 November. The optical spectrum of V1 is dominated by H emission lines, with Fe II, He I and Na I also detected. In the ultraviolet, a forest of Fe absorption features and numerous absorption lines typical of mid-B supergiants are observed. From a spectral analysis with the non-LTE, line-blanketed code of Hillier & Miller (1998), we derive stellar parameters of T*=11kK, R*=420Ro, log(L/Lo)=6.35 during 1997 November, and T*=13kK, R*=315Ro, log(L/Lo)=6.4 for 1999 July. The wind properties of V1 are also exceptional, with Mdot ~ 4.4 x 10e-4 Mo/yr and $v_{∞} ~ $300 km/s, allowing for a clumped wind (filling factor = 0.3), and assuming H/He ~ 4 by number. The presence of Fe lines in the UV and optical spectrum of V1 permits an estimate of the heavy elemental abundance of NGC 2363 from our spectral synthesis. Although some deficiencies remain, allowance for charge exchange reactions in our calculations supports a SMC-like metallicity, that has previously been determined for NGC 2363 from nebular oxygen diagnostics. Considering a variety of possible progenitor stars, V1 has definitely undergone a giant eruption, with a substantial increase in stellar luminosity, radius, and almost certainly mass-loss rate. ; Comment: 29 pages, 10 figures, Submitted to ApJ