The General Antiparticle Spectrometer (GAPS) is an upcoming balloon mission to measure low-energy cosmic-ray antinuclei during at least three ∼35-day Antarctic flights. With its large geometric acceptance and novel exotic atom-based particle identification, GAPS will detect ∼500 cosmic antiprotons per flight and produce a precision cosmic antiproton spectrum in the kinetic energy range of ∼0.07−0.21GeV/n at the top of the atmosphere. With these high statistics extending to lower energies than any previous experiment, and with complementary sources of experimental uncertainty compared to traditional magnetic spectrometers, the GAPS antiproton measurement will be sensitive to dark matter, primordial black holes, and cosmic ray propagation. The antiproton measurement will also validate the GAPS antinucleus identification technique for the antideuteron and antihelium rare-event searches. This analysis demonstrates the GAPS sensitivity to cosmic-ray antiprotons using a full instrument simulation and event reconstruction, and including solar and atmospheric effects.