We report the discovery of diffuse, highly ionized line emission from the λ1550 lines of C IV and the λ1663 lines of O III, observed by the Berkeley EUV/FUV Shuttle Telescope on Shuttle mission 61-C. The C IV line was detected in four directions at 5 σ, 6 σ, 8 σ, and 3 σ significance. The O III] line was detected at 3 σ in two out of the eight directions searched. There are also detections at greater than 95% confidence of O IV/Si IV λ1401 and N III] λ1750 in the summed high-latitude spectrum. The C IV and the O III] line intensities exhibit an anticorrelation with H I column density. While the emission-line ratios are consistent with a single temperature in most directions, there is some evidence that the emission is the result of gas with a nonequilibrium ionization balance and a with a distribution of temperatures. The density and pressure of the emission line gas can be derived from a comparison with published C IV absorption column densities. If we assume that the emission is produced by all of the absorbing gas, the density is 0.01 cm-3, and the pressure p/k ≃ 1000 cm-3. This is consistent with other estimates of gas pressure at 2-3 kpc from the plane, while it is 5-10 times lower than the disk pressure. If the gas is local (100-200 pc), our observations imply that the gas pressure is 5-10 times greater than the disk pressure. Furthermore, local gas is unlikely to produce the observed anticorrelation with H I column density. We show that photoionized galactic halo models cannot account for this emission. Of the models proposed to explain the presence of highly ionized gas in the local region of the Galaxy, we conclude that our observation of UV emission is consistent only with a galactic fountain model. This fountain has a mass flux of 6-25 M⊙ yr-1 and a luminosity of 2.5-13 × 1040 ergs s-1.