We present a high-resolution spectrum (lambda/Delta lambda = 3000) of the Small Magellanic Cloud (SMC) star NGC 346 No. 1 in the vicinity of the O vr resonance doublet (1031.9, 1037.6 Angstrom). The spectrum was measured with the Berkeley EUV/FUV spectrometer in the ORFEUS telescope aboard the space platform Astro-SPAS during the mission of space shuttle Discovery flown in September of 1993. This instrument is the first to observe extragalactic targets in the far-ultraviolet with sufficient spectral resolution to clearly separate the O VI from other species that provide strong absorption at nearby wavelengths. We find an O VI feature centered near nu(LSR) = + 140 km s(-1). This is almost certainly associated with the OB cluster NGC 346 itself, which also shows deep absorption at Si IV and C IV (but little N V) at comparable velocities. An absorption dip corresponding to O VI near nu(LSR) approximate to -230 km s(-1) is present in the data, but this may harbor a contribution from features of other stars within the 20'' circular aperture, or from the wind of NGC 346 No. 1 itself. Between these velocity extrema lies either one broad (b approximate to 70 km s(-1)) or multiple narrower features with an apparent column density of about 1.3 x 10(14) cm(-2); however, this may also include contributions from phenomena unrelated to interstellar O vr. Taking the sum of the inferred column densities of both the high negative velocity and low-velocity O vr we arrive at 2 x 10(14) cm(-2) as an upper limit to Galactic halo O VI toward NGC 346 No. 1. Assuming an exponential distribution of O VI with the in-plane density of Jenkins (1978b), we infer an upper limit to the scale height, h(0), of about 1.7 kpc toward NGC 346 No. 1. We revisit the ORFEUS spectrum of PKS 2155-304, initially presented in Appenzeller et al. (1995), and find a 2 sigma upper limit to h(0) of 2.0 kpc along this sight line. The upper limits to the O vr column densities toward the SMC and PKS 2155-304 are a factor of similar to 4 below the predictions of a cooling fountain model (Edgar and Chevalier 1986, as parameterized by Sembach, Savage, and Massa 1991). The low O VI scale height is consistent with the recent predictions of Shull and Slavin (1994).