The C IV doublet in the z = 1.795 absorption-line system in the QSO B2 1225 + 317 has been observed with the Multiple Mirror Telescope echelle spectrograph at a resolution of 10 km/s. The profile of C IV, previously known to contain at least three velocity components, is shown to consist of at least eleven. Individual components have Doppler width upper limits of 6.4-13.2 km/s. These low upper limits, plus a study of published profile shapes for several other species, suggest that the observed gas is not in an equilibrium state of collisional ionization, and that electron temperatures are 35,000 K or lower. The width of the aggregate profile (approximately 500 km/s) is larger by a factor of 5 than any C IV or Si IV profile observed through the halo of the Galaxy, while individual components are narrower than those in the Galaxy. Thus, intervening galaxies with halos similar to that of the Galaxy, even if occurring in groups, may not explain this QSO absorption-line system.