We present the most sensitive ultraviolet observations of Supernova 1987A to date. Imaging spectroscopy from the Hubble Space Telescope-Cosmic Origins Spectrograph shows many narrow (dv sim 300 km/s) emission lines from the circumstellar ring, broad (dv sim 10 -- 20 x 10^3 km/s) emission lines from the reverse shock, and ultraviolet continuum emission. The high signal-to-noise (> 40 per resolution element) broad LyA emission is excited by soft X-ray and EUV heating of mostly neutral gas in the circumstellar ring and outer supernova debris. The ultraviolet continuum at lambda > 1350A can be explained by HI 2-photon emission from the same region. We confirm our earlier, tentative detection of NV lambda 1240 emission from the reverse shock and we present the first detections of broad HeII lambda1640, CIV lambda1550, and NIV] lambda1486 emission lines from the reverse shock. The helium abundance in the high-velocity material is He/H = 0.14 +/- 0.06. The NV/H-alpha line ratio requires partial ion-electron equilibration (T_{e}/T_{p} approx 0.14 - 0.35). We find that the N/C abundance ratio in the gas crossing the reverse shock is significantly higher than that in the circumstellar ring, a result that may be attributed to chemical stratification in the outer envelope of the supernova progenitor. The N/C abundance ratio may have been stratified prior to the ring expulsion, or this result may indicate continued CNO processing in the progenitor subsequent to the expulsion of the circumstellar ring.
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