To study changes in the gas-phase abundances of elements in the Galactic halo, we have used the Goddard High Resolution Spectrograph instrument on the Hubble Space Telescope to observe interstellar absorption lines of O I, Mg II, Al II, Al III, Si II, S II, Fe II, and Ni II in the spectra of three stars at high Galactic latitudes: HD 22586, HD 49798, and HD 120086. Recognizing that we would probably observe a decrease in the amount of depletions of certain heavy elements onto dust grains, we also made comparison observations of HD 72089 behind the Vela supernova remnant. This star has a number of high-velocity gas components that have been subjected to strong shocks, allowing us to determine whether the patterns of abundances, in a medium where the grains have been partially destroyed, had any resemblance to abundances in the halo. When we compare various element abundances relative to that of sulfur, a normally undepleted element, we find that as a rule the element depletions of gas toward the halo stars are intermediate between high- and low-velocity gas in the spectrum of HD 72089. Exceptions to this are the extraordinarily light depletions of Si toward HD 22586 and HD 120086 (-0.1 and 0.0 dex, respectively), and a very mild depletion of Mg in HD 120086 (-0.2 dex). We see no evidence that iron-peak elements are enhanced relative to the others (beyond the effects of changes in depletion), as one might expect if the ejecta from Type Ia supernovae were enriching the gas in the halo. It therefore seems apparent that either there is not enough change in the Type II supernovae to Type Ia supernovae ratio from the plane to the halo to produce a noticeable enrichment, or else one or more gas transport processes between the disk and halo, such as mass exchange in a Galactic fountain or the general turbulence of the interstellar medium, are sufficiently strong to cycle material between the disk and halo at a rate that can stifle any abundance contrasts.