  We show that the existence of a sub-dominant form of dark matter, made of
dark antiatoms of mass and size of the order of 1 TeV and 30 fm respectively,
can explain the results of direct detection experiments, with a positive signal
in DAMA/NaI and DAMA/LIBRA and no signal in other experiments. The signal comes
from the binding of the dark antiatoms to thallium, a dopant in DAMA, and is
not present for the constituent atoms of other experiments. The dark antiatoms
are made of two particles oppositely charged under a dark U(1) symmetry and can
bind to terrestrial atoms because of a kinetic mixing between the photon and
the massless dark photon, such that the dark particles acquire an electric
millicharge of the order of 0.0005e. This millicharge enables them to bind to
high-Z atoms via radiative capture, after they thermalize in terrestrial matter
through elastic collisions.
