  If a component of the dark matter has dissipative interactions, it could
collapse to form a thin dark disk in our Galaxy that is coplanar with the
baryonic disk. It has been suggested that dark disks could explain a variety of
observed phenomena, including periodic comet impacts. Using the first data
release from the $Gaia$ space observatory, we search for a dark disk via its
effect on stellar kinematics in the Milky Way. Our new limits disfavor the
presence of a thin dark matter disk, and we present updated measurements on the
total matter density in the Solar neighborhood.
