  We prove global well-posedness below the charge norm (i.e., the $L^2$ norm of
the Dirac spinor) for the Dirac-Klein-Gordon system of equations (DKG) in one
space dimension. Adapting a method due to Bourgain, we split off the high
frequency part of the initial data for the spinor, and exploit nonlinear
smoothing effects to control the evolution of the high frequency part. To prove
the nonlinear smoothing we rely on the null structure of the DKG system, and
bilinear estimates in Bourgain-Klainerman-Machedon spaces.
