We report the discovery of a Milky Way satellite in the constellation of Antlia. The Antlia 2 dwarf galaxy is located behind the Galactic disc at a latitude of b~11{deg} and spans 1.26{deg}, which corresponds to ~2.9kpc at its distance of 130kpc. While similar in spatial extent to the Large Magellanic Cloud, Antlia 2 is orders of magnitude fainter at M_V_=-9mag, making it by far the lowest surface brightness system known (at ~31.9mag/arcsec^2^), ~100 times more diffuse than the so-called ultra diffuse galaxies. The satellite was identified using a combination of astrometry, photometry, and variability data from Gaia Data Release 2, and its nature confirmed with deep archival DECam imaging, which revealed a conspicuous BHB signal. We have also obtained follow-up spectroscopy using AAOmega on the AAT, identifying 159 member stars, and we used them to measure the dwarf's systemic velocity, 290.9+/-0.5km/s, its velocity dispersion, 5.7+/-1.1km/s, and mean metallicity, [Fe/H]=-1.4. From these properties we conclude that Antlia 2 inhabits one of the least dense dark matter (DM) haloes probed to date. Dynamical modelling and tidal-disruption simulations suggest that a combination of a cored DM profile and strong tidal stripping may explain the observed properties of this satellite. The origin of this core may be consistent with aggressive feedback, or may even require alternatives to cold dark matter (such as ultra-light bosons).