We present ultraviolet interstellar absorption-line measurements for the sightline toward the O9.5 V star mu Columbae (l = 237.degrees 3, b = -27.degrees 1; d approximate to 400 pc, z approximate to 180 pc; [n(H I)] approximate to 0.06 cm(-3)) obtained with the Goddard High Resolution Spectrograph (GHRS) on board the Hubble Space Telescope. These archival data represent the most complete GHRS interstellar absorption-line measurements for any line of sight toward an early-type star. The 3.5 km s(-1) resolution of the instrument allows us to accurately derive the gas-phase column densities of many important ionic species in the diffuse warm neutral medium, including accounting for saturation effects in the data and for contamination from ionized gas along this sightline. For the low-velocity material (-20 less than or similar to v(LSR) less than or similar to +15 km s(-1)), we use the apparent column density method to derive column densities. For the individual absorbing components at v(LSR) approximate to -28.8, +20.1, +31.0, and +41.2 km s(-1), we apply component fitting techniques to derive column densities and b-values. We have also used observations of interstellar Ly alpha absorption taken with the GHRS intermediate resolution gratings to accurately derive the H I column density along this sightline. The resulting interstellar column density, log N(H I) = 19.86 +/- 0.015, is in agreement with other determinations but is significantly more precise. The low-velocity material shows gas-phase abundance patterns similar to the warm cloud (cloud A) toward the disk star zeta Ophiuchi, while the component at v(LSR) approximate to +20.1 km s(-1) shows gas-phase abundances similar to those found in warm halo clouds. We find that the velocity-integrated gas-phase abundances of Zn, P, and S relative to H along this sightline are indistinguishable from solar system abundances. We discuss the implications of our gas-phase abundance measurements for the composition of interstellar dust grains. We find a dust-phase abundance [(Fe + Mg)/Si](d) = 2.7-3.3 in the low-velocity gas; therefore the dust cannot be composed solely of common silicate grains, but must also include oxides or pure iron grains. The low-velocity material along this sightline is characterized by T approximate to 6000-7000 K with n(e) approximate to 0.3 cm(-3), derived from the ionization equilibrium of Mg and Ca. The relative ionic column density ratios of the intermediate-velocity components at v(LSR) = +31.0 and +41.2 km s(-1) show the imprint both of elemental incorporation into grains and (photo)ionization. These clouds have low total hydrogen column densities [log N(H) similar to 17.4-17.7], and our component fitting b-values constrain the temperature in the highest velocity component to be T = 4000 + 700 K. The electron density of this cloud is n(e) approximate to 0.6 cm(-3), derived from the P-2(1/2) to P-2(3/2) fine structure excitation of C II. The components at v(LSR) approximate to -30 and -48 km s(-1) along this sightline likely trace shocked gas with very low hydrogen column densities. The v(LSR) approximate to -30 km s(-1) component is detected in a few strong low-ionization lines while both are easily detected in Si III. The relative column densities of the -30 km s(-1) suggest that the gas is collisionally ionized at moderate temperatures (T approximate to 25,000 K). This is consistent with the measured b-values of this component, though nonthermal motions likely contribute significantly to the observed breadths.