We have searched for far-UV emission from coronal gas in the Galactic halo along lines of sight with Galactic latitudes between 42 degrees and 88 degrees in spectra obtained by the Hopkins Ultraviolet Telescope during the Astro-2 space shuttle mission in 1995 March. Of four spectra taken through a 19 '' x 197 '' aperture, two show O VI lambda lambda 1032, 1038 emission at a significance of 4 sigma. Two out of six spectra taken through a 10 '' x 56 '' aperture also show O VI emission, though at only 2 sigma levels of significance. Three of the detections lie near regions of enhanced soft X-ray emission associated with Radio Loop I. The fourth, at l = 218 degrees, b = 56 degrees, may represent a more typical region of the halo. In its spectrum, we find I(O VI)= (3.59 +/- 0.96) x 10(-7) ergs cm(-2) s(-1). This is the first detection of O VI emission from the Galactic halo. None of the spectra exhibit significant emission from C IV lambda lambda 1548, 1551, though our upper limits are greater than the intensities reported for other lines of sight. We set a limit on I(O VI)/I(C IV) greater than or equal to 3.4, consistent with the predictions of self-photoionizing Galactic fountain models, but higher than those of models based on turbulent mixing layers. Combining our measured O VI intensity with estimates of N(O VI) through the halo, we find that, for 5.3 less than or equal to log T less than or equal to 5.8, the data are consistent with n(e) approximate to 0.06 cm(-3) and 22,000 less than or equal to P/k less than or equal to 67,000 cm(-3) K, values substantially greater than those derived from C IV observations, suggesting that the C IV and O VI emission arise from physically distinct clouds and/or that a substantial portion of the C IV absorption arises from cooler gas that does not contribute to the C IV emission. This result is consistent with Galactic halo models incorporating self-photoionization of the cooling gas.