Using the Hopkins Ultraviolet Telescope on the Astro-1 space shuttle mission in 1990 December, we have obtained a far-ultraviolet spectrum (912-1860 angstrom at 3.5 angstrom resolution) of a complex of bright filaments located in the eastern part of the supernova remnant N49 in the Large Magellanic Cloud. This is the first observation of an extragalactic supernova remnant that extends to the Lyman limit. We detect lines of O VI lambda1035, O IV] lambda1403, C IV lambda1550, and He II lambda1640. The O VI emission that we observe cannot originate in shocks with velocities less-than-or-equal-to 140 km s-1 which are responsible for the bulk of both the C IV and the optical line emission. Likewise, the main blast wave with a velocity approximately 730 km s-1 is unable to account for the brightness of O VI. Most of the O VI originates in optically faint, 190-270 km s-1 shocks, traversing clouds with densities of 20-40 cm-3. We expand upon earlier modeling of N49 IUE and optical spectra that assumed a distribution of shock velocities; including data from below Lyalpha, such models still describe the data well.