  The search for weakly interacting massive particles (WIMPs) by direct
detection faces an encroaching background due to coherent neutrino-nucleus
scattering. As the sensitivity of these experiments improves, the question of
how to best distinguish a dark matter signal from neutrinos will become
increasingly important. A proposed method of overcoming this so-called
'neutrino floor' is to utilize the directional signature that both neutrino and
dark matter induced recoils possess. We show that directional experiments can
indeed probe WIMP-nucleon cross-sections below the neutrino floor with little
loss in sensitivity due to the neutrino background. In particular we find at
low WIMP masses (around 6 GeV) the discovery limits for directional detectors
penetrate below the non-directional limit by several orders of magnitude. For
high WIMP masses (around 100 GeV), the non-directional limit is overcome by a
factor of a few. Furthermore we show that even for directional detectors which
can only measure 1- or 2-dimensional projections of the 3-dimensional recoil
track, the discovery potential is only reduced by a factor of 3 at most. We
also demonstrate that while the experimental limitations of directional
detectors, such as sense recognition and finite angular resolution, have a
detrimental effect on the discovery limits, it is still possible to overcome
the ultimate neutrino background faced by non-directional detectors.
