  We show that for planar dispersing billiards the return times distribution
is, in the limit, Poisson for metric balls almost everywhere w.r.t. the SRB
measure. Since the Poincar\'e return map is piecewise smooth but becomes
singular at the boundaries of the partition elements, recent results on the
limiting distribution of return times cannot be applied as they require the
maps to have bounded second derivatives everywhere. We first prove the Poisson
limiting distribution assuming exponentially decaying correlations. For the
case when the correlations decay polynomially, we induce on a subset on which
the induced map has exponentially decaying correlations. We then prove a
general theorem according to which the limiting return times statistics of the
original map and the induced map are the same.
