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 $1.3^{+4.5}_{-1.2}cdot 10^{-6}mathrm{m^{-2}sr^{-1}s^{-1}}(mathrm{GeV}/n)^{-1}$ (95% confidence level) in the energy range of 0.11$-$0.3 GeV/$n$, opening a new window on rare cosmic physics.
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