We report the first Far Ultraviolet Spectroscopic Explorer (FUSE) measurements of diffuse O VI (lambda,lambda 1032,1038) emission from the general diffuse interstellar medium outside of supernova remnants or superbubbles. We observed a 30arcsec x 30arcsec region of the sky centered at l = 315 and b = -41. From the observed intensities (2930+/-290(random)+/-410(systematic) and 1790+/-260(random)+/-250(systematic) photons/cm/cm/s/sr in the 1032 and 1038 Angstrom emission lines, respectively), derived equations, and assumptions about the source location, we calculate the intrinsic intensity, electron density, thermal pressure, and emitting depth. The intensities are too large for the emission to originate solely in the Local Bubble. Thus, we conclude that the Galactic thick disk and lower halo also contribute. High velocity clouds are ruled out because there are none near the pointing direction. The calculated emitting depth is small, indicating that the O VI-bearing gas fills a small volume. The observations can also be used to estimate the cooling rate of the hot interstellar medium and constrain models. The data also yield the first intensity measurement of the C II 3s2 S1/2 to 2p2 P3/2 emission line at 1037 Angstroms and place upper limits on the intensities of ultraviolet line emission from C I, C III, Si II, S III, S IV, S VI, and Fe III.