  Local well-posedness for the Dirac - Klein - Gordon equations is proven in
one space dimension, where the Dirac part belongs to H^{-{1/4}+\epsilon} and
the Klein - Gordon part to H^{{1/4}-\epsilon} for 0 < \epsilon < 1/4, and
global well-posedness, if the Dirac part belongs to the charge class L^2 and
the Klein - Gordon part to H^k with 0 < k < 1/2 . The proof uses a null
structure in both nonlinearities detected by d'Ancona, Foschi and Selberg and
bilinear estimates in spaces of Bourgain-Klainerman-Machedon type.
