We describe an ultraviolet spectroscopic survey of interstellar high-velocity cloud (HVC) absorption in the strong λ1206.500 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 λ1031.926, it provides a sensitive probe of ionized gas down to column densities N Si III 5 × 1011 cm-2 at Si III equivalent width 10 m. We detect high-velocity Si III over 91% 4% of the sky (53 of 58 sight lines); 59% of the HVCs show negative local standard of rest velocities. The mean HVC column density per sight line is 〈log N Si III 〉 = 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 104.0-4.5 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 1018-1019 cm-2 for gas of hydrogen density n H 0.1 cm-3 and 10% solar metallicity. With typical neutral fractions N H I/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 8M and produce a mass infall rate of 1 M yr-1 to the Galactic disk.