We confirm a massive sub-Neptune-sized planet on a P=22.8 days orbit around the star TOI-1824 (Teff=5200 K, V=9.7 mag). TESS first identified TOI-1824 b (formerly TOI-1824.01) as an object of interest in 2020 April after two transits in Sector 22 were matched with a single transit in Sector 21. TOI-1824 was subsequently targeted for ground-based Doppler monitoring with Keck-HIRES and APF-Levy. Using a joint model of the TESS photometry, radial velocities, and Ca ii H and K emission measurements as an activity indicator, we find that TOI-1824b is an unusually dense sub-Neptune. The planet has a radius Rp=2.63{+/-}0.15R{Earth} and mass Mp=18.5{+/-}3.2M{Earth}, implying a bulk density of 5.6{+/-}1.4g/cm^3^. TOI-1824 b's mass and radius situate it near a small group of "superdense sub-Neptunes" (Rp<~3R{Earth} and Mp>~20M{Earth}). While the formation mechanism of superdense sub-Neptunes is a mystery, one possible explanation is the constructive collision of primordial icy cores; such giant impacts would drive atmospheric escape and could help explain these planets' apparent lack of massive envelopes. We discuss TOI-1824b in the context of these overdense planets, whose unique location in the exoplanet mass-radius plane make them a potentially valuable tracer of planet formation.