We report the discovery of a nearby dwarf galaxy in the constellation of Hydrus, between the Large (LMC) and the Small Magellanic Clouds (SMC). Hydrus 1 is a mildly elliptical ultrafaint system with luminosity MV ∼ −4.7 and size 53 ± 3 pc, located 28 kpc from the Sun and 24 kpc from the LMC. From spectroscopy of ∼30 member stars, we measure a velocity dispersion of 2.7 ± 0.5 km s−1 and find tentative evidence for a radial velocity gradient consistent with 3 km s−1 rotation. Hydrus 1’s velocity dispersion indicates that the system is dark matter dominated, but its dynamical mass-to-light ratio M/L=66+29−20 is significantly smaller than typical for ultrafaint dwarfs at similar luminosity. The kinematics and spatial position of Hydrus 1 make it a very plausible member of the family of satellites brought into the Milky Way by the Magellanic Clouds. While Hydrus 1’s proximity and well-measured kinematics make it a promising target for dark matter annihilation searches, we find no evidence for significant gamma-ray emission from Hydrus 1. The new dwarf is a metal-poor galaxy with a mean metallicity [Fe/H]=−2.5 and [Fe/H] standard deviation of 0.4 dex, similar to other systems of similar luminosity. α abundances of Hyi 1 members indicate that star formation was extended, lasting between 0.1 and 1 Gyr, with self-enrichment dominated by supernova Ia. The dwarf also hosts a highly carbon-enhanced extremely metal-poor star with [Fe/H]∼−3.2 and [C/Fe] ∼ +3.0.