The EXoplanet Climate Infrared TElescope (EXCITE) experiment is a balloon-borne, purpose-designed mission to measure spectroscopic phase curves of short-period extrasolar giant planets (EGPs, or "hot Jupiters"). Here, we present EXCITE's principal science instrument: a high-throughput, single-object spectrograph operating in the 0.8-2.5 mu m and 2.5-4.0 mu m bands with R >= 50. Our compact design achieves diffraction-limited, on-axis performance with just three powered optics: two off-axis parabolic mirrors and a CaF2 prism. We discuss the optical and mechanical design, the expected optical performance of the spectrograph, and summarize the tolerances needed to achieve that performance. We also discuss plans for establishing alignment of the optics and verifying the optical performance.