  Making robust predictions for the phase space distribution of dark matter at
the solar neighbourhood is vital for dark matter direct detection experiments.
To date, almost all such predictions have been based on simulations that model
the dark matter alone. Here, we use three cosmological hydrodynamics
simulations of bright, disc dominated galaxies to include the effects of
baryonic matter self-consistently for the first time. We find that the addition
of baryonic physics drastically alters the dark matter profile in the vicinity
of the Solar neighbourhood. A stellar/gas disc, already in place at high
redshift, causes merging satellites to be dragged preferentially towards the
disc plane where they are torn apart by tides. This results in an accreted dark
matter disc that contributes ~0.25 - 1.5 times the non-rotating halo density at
the solar position. The dark disc, unlike dark matter streams, is an
equilibrium structure that must exist in disc galaxies that form in a
hierarchical cosmology. Its low rotation lag with respect to the Earth
significantly boosts WIMP capture in the Earth and Sun, boosts the annual
modulation signal, and leads to distinct variations in the flux as a function
of recoil energy that allow the WIMP mass to be determined.
