We report the discovery of a nearby dwarf galaxy in the constellation of Hydrus, between the Large and the Small Magellanic Clouds. Hydrus 1 is a mildy elliptical ultra-faint system with luminosity $M_Vsim$ -4.7 and size $sim$ 50 pc, located 28 kpc from the Sun and 24 kpc from the LMC. From spectroscopy of $sim$ 30 member stars, we measure a velocity dispersion of 2.7 km/s and find tentative evidence for a radial velocity gradient consistent with 3 km/s rotation. Hydrus 1s velocity dispersion indicates that the system is dark matter dominated, but its dynamical mass-to-light ratio M/L $sim$ 66 is significantly smaller than typical for ultra-faint 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 1s 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] spread of 0.4 dex, similar to other systems of similar luminosity. Alpha-abundances of Hyi 1 members indicate that star-formation was extended, lasting between 0.1 and 1 Gyr, with self-enrichment dominated by SN Ia. The dwarf also hosts a highly carbon-enhanced extremely metal-poor star with [Fe/H] $sim$ -3.2 and [C/Fe] $sim$ +3.0.
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