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 M_V_~-4.7 and size 53+/-3pc, located 28kpc from the Sun and 24kpc from the LMC. From spectroscopy of ~30 member stars, we measure a velocity dispersion of 2.7+/-0.5 km/s and find tentative evidence for a radial velocity gradient consistent with 3km/s 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 0.4dex, similar to other systems of similar luminosity. {alpha} abundances of Hyi 1 members indicate that star formation was extended, lasting between 0.1 and 1Gyr, 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.