We present deep ultraviolet observations of a field containing the cluster Abell 2246 (z=0.225) which provide far-ultraviolet (FUV) images of some of the faintest galaxies yet observed in that bandpass. Abell 2246 lies within the field of view of Ultraviolet Imaging Telescope (UIT) observations of the quasar HS1700+64, which accumulated over 7100 seconds of UIT FUV exposure time during the Astro-2 mission in March 1995. For objects found on both the FUV and ground-based V-band images, we obtain FUV (l ~ 1520 A) photometry and V-band photometry, as well as mid-UV (l ~ 2490 A) photometry from UIT Astro-1 observations and ground-based I-band photometry. We find five objects in the images which are probably galaxies at the distance of Abell 2246, with FUV magnitudes (m(FUV)) between 18.6 and 19.6, and V magnitudes between 18.4 and 19.6. We find that their absolute FUV fluxes and colors imply strongly that they are luminous galaxies with significant current star formation, as well as some relatively recent, but not current, (> 400 Myr ago) star formation. We interpret the colors of these five objects by comparing them with local objects, redshift-corrected template spectra and stellar population models, finding that they are plausibly matched by 10-Gyr-old population models with decaying star formation, with decay time constants in the range 3 Gyr < t < 5 Gyr, with an additional color component from a single burst of moderate ( ~ 400-500 Myr) age. From derived FUV luminosities we compute current star formation rates. We compare the UV properties of Abell 2246 with those of the Coma cluster, finding that Abell 2246 has significantly more recent star formation, consistent with the Butcher-Oemler phenomenon.