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Goddard High Resolution Spectrograph echelle-mode observations of the interstellar absorption lines of Mg II, Si IV, C IV, and N V toward µ Columbae (HD 38666) are presented. The observations have a spectral resolution of 3.5 km.s^–1 and signal-to-noise ratios (S/Ns) of 20-200. The µ Col sight line (l = 237°.3, b = -27°.1, d=0.40 kpc, z = -0.18 kpc) extends though the Local Bubble and the warm neutral, warm ionized, and hot ionized phases of the interstellar medium (ISM). The high-ionization column densities toward µ Col are log N(Si IV) = 12.16±0.05, log N(C IV) = 12.88±0.02, and logN(N V) = 11.8-12.3. Profile fits to Copernicus satellite measures of O VI absorption toward µ Col yield log N(O VI) = 13.82±0.01 and b = 38.7 km.s^–1. This implies N(C IV)/N(O VI) = 0.11±0.01, which is typical of the values found for the hot ISM of the Galactic disk. The O VI profile is twice as broad as the C IV and N V profiles, even though these species have roughly similar average velocities. Some of the C IV, N V, and O VI absorption toward µ Col may occur at the interface of the Local Cloud and Local Bubble, although additional contributions to these ions probably also occur in more distant gas along the sight line. A substantial part of the Si IV absorption likely arises in warm photoionized gas in an H II region surrounding µ Col. The profile width differences among the high-ionization lines of C IV, N V, and O VI could be produced if the line of sight passes through a highly evolved supernova remnant. The observations for µ Col and for other stars observed at high resolution with the GHRS reveal that multiple gas types (warm and hot) contribute to the absorption by the highly ionized atoms along both nearby and distant sight lines. Disentangling the relative contributions from the different gas types requires high-resolution and high-S/N observations. The Mg II observations, combined with a solar Mg reference abundance, imply that the Mg depletion toward µ Col is -0.31 dex. As observed for other sight lines through the warm neutral medium, the gas-phase observations of Mg, when combined with results for Fe and Si, suggest that Mg and Fe are more deficient from the gas phase than one would expect if these elements are only contained in silicate dust grains.Key words: