The General Antiparticle Spectrometer (GAPS) is an Antarctic balloon experiment designed for low-energy (0.1–0.3 GeV/n) cosmic antinuclei as signatures of dark matter annihilation or decay. GAPS is optimized to detect low-energy antideuterons, as well as to provide unprecedented sensitivity to low-energy antiprotons and antihelium nuclei. The novel GAPS antiparticle detection technique, based on the formation, decay, and annihilation of exotic atoms, provides greater identification power for these low-energy antinuclei than previous magnetic spectrometer experiments. This work reports the sensitivity of GAPS to detect antihelium-3 nuclei, based on full instrument simulation, event reconstruction, and realistic atmospheric influence simulations. The report of antihelium nuclei candidate events by AMS-02 has generated considerable interest in antihelium nuclei as probes of dark matter and other beyond the Standard Model theories. GAPS is in a unique position to detect or set upper limits on the cosmic antihelium nuclei flux in an energy range that is essentially free of astrophysical background. In three 35-day long-duration balloon flights, GAPS will be sensitive to an antihelium flux on the level of 1.3−1.2+4.5·10−6 m-2sr-1s-1(GeV/n)-1 (95% confidence level) in the energy range of 0.11–0.3 GeV/n, opening a new window on rare cosmic physics.