The giant elliptical galaxy NGC 1399 was observed with the Hopkins Ultraviolet Telescope (HUT) aboard the Astro-1 space shuttle mission in 1990 December. These are the first data to provide measurements of the spectral energy distribution of an elliptical galaxy shortward of Ly-alpha and to allow detection of absorption features in the sub 2000 angstrom UV. The spectrum shows no emission features other than well-known geocoronal lines. Strong Lyman series absorption is evident at the galaxy redshift. Other absorption features attributable to the hot stellar population are detected, but there is no evidence of C IV lambda-lambda-1548, 1551 absorption. We use these data to set constraints on the origin of the UV upturn in elliptical galaxies. The lack of detectable C IV absorption and the shape of the continuum spectrum exclude star formation with a standard IMF as the sole contributor to the 1550 angstrom flux. The spectral energy distributions predicted by the "hot-E" model of Guiderdoni and Rocca-Volmerange and the Bruzual mu-models do not match the observed spectrum. Postasymptotic giant branch (PAGB) stars can account for the spectrum near 1550 angstrom, because many have low carbon abundance in their envelopes. However, synthetic spectra produced from theoretical evolutionary tracks and stellar model atmospheres fail to fit to the observed spectral-energy distribution. If PAGB stars produce the UV upturn, then either they do not evolve along the standard tracks, or the (solar metallicity) model atmospheres used to construct the synthetic spectrum are inappropriate. The observed continuum flux decreases from 1050 angstrom to the Lyman limit, indicating that the light is dominated by stars with temperatures less than 25,000 K. For star-formation models, this temperature puts the main-sequence turnoff near B0, implying either that star formation ended approximately 2 x 10(7) yr ago, or that stars are currently forming with a truncated IMF. The continuum shape and presence of Lyman-series absorption suggest extreme horizontal-branch star models deserve additional attention.