We describe an ultraviolet spectroscopic survey of interstellar high-velocity cloud (HVC) absorption in the strong 1206.500 Angstrom line of Si III using the Space Telescope Imaging Spectrograph aboard the Hubble Space Telescope. Because the Si III line is 4-5 times stronger than O VI 1031.926, it provides a sensitive probe of ionized gas down to column densities N(Si III) = 5x10^11 cm^-2 at Si III equivalent width 10 mA. We detect high-velocity Si III over (91+/-4)% of the sky (53 of 58 sight lines), and 59% of the HVCs show negative LSR velocities. Per sight line, the mean HVC column density is <log N(SiIII)> = 13.19 +/- 0.45, while the mean for all 90 velocity components is 12.92 +/- 0.46. Lower limits due to Si III line saturation are included in this average, so the actual mean/median values are even higher. The Si III appears to trace an extensive ionized component of Galactic halo gas at temperatures 10^4.0 K to 10^4.5 K indicative of a cooling flow. Photoionization models suggest that typical Si III absorbers with 12.5 < log N(Si III) < 13.5 have total hydrogen column densities N(H) = 10^18 to 10^19 cm^-2 for gas of hydrogen density n(H) = 0.1 cm^(-3) and 10% solar metallicity. With typical neutral fractions N(HI)/N(H) = 0.01, these HVCs may elude even long duration 21-cm observations at Arecibo, the EVLA, and other radio facilities. However, if Si III is associated with higher density gas, n(H) > 1 cm^(-3), the corresponding neutral hydrogen could be visible in deep observations. This reservoir of ionized gas may contain 10^8 M_sun and produce a mass infall rate of 1 M_sun/yr to the Galactic disk.
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