  We extract at redshift z=0 a Milky Way sized object including gas, stars and
dark matter (DM) from a recent, high-resolution cosmological N-body simulation
with baryons. Its resolution is sufficient to witness the formation of a
rotating disk and bulge at the center of the halo potential. The phase-space
structure of the central galactic halo reveals the presence of a dark disk
component, that is co-rotating with the stellar disk. At the Earth's location,
it contributes to around 25% of the total DM local density, whose value is
rho_DM ~ 0.37 GeV/cm^3. The velocity distributions also show strong deviations
from pure Gaussian and Maxwellian distributions, with a sharper drop of the
high velocity tail.
  We give a detailed study of the impact of these features on the predictions
for DM signals in direct detection experiments. In particular, the question of
whether the modulation signal observed by DAMA is or is not excluded by limits
set by other experiments (CDMS, XENON and CRESST...) is re-analyzed and
compared to the case of a standard Maxwellian halo, in both the elastic and the
inelastic scattering scenarios. We find that the compatibility between DAMA and
the other experiments is improved. In the elastic scenario, the DAMA modulation
signal is slightly enhanced in the so-called channeling region, as a result of
several effects. For the inelastic scenario, the improvement of the fit is
mainly attributable to the departure from a Maxwellian distribution at high
velocity.
