New measurements with good statistics will make it possible to observe the time variation of cosmic antiprotons at I AU through the approaching peak of solar activity. We report a new computation of the interstellar antiproton spectrum expected from collisions between cosmic protons and the interstellar gas. This spectrum is then used as input to a steady-state drift model of solar modulation, in order to provide predictions for the antiproton spectrum as well as the antiproton/proton ratio at 1 AU. Our model predicts a surprisingly large, rapid increase in the antiproton/proton ratio through the next solar maximum, followed by a large excursion in the ratio during the following decade.