Without a source of new gas, our Galaxy would exhaust its supply of gas through the formation of stars. Ionized gas clouds observed at high velocity may be a reservoir of such gas, but their distances are key for placing them in the galactic halo and unraveling their role. We have used the Hubble Space Telescope to blindly search for ionized high-velocity clouds (iHVCs) in the foreground of galactic stars. We show that iHVCs with 90 ≤ |v(LSR)| ≲ 170 kilometers per second (where v(LSR) is the velocity in the local standard of rest frame) are within one galactic radius of the Sun and have enough mass to maintain star formation, whereas iHVCs with |v(LSR)| ≳ 170 kilometers per second are at larger distances. These may be the next wave of infalling material.