  We propose a differential difference equation in ${\mathcal R}^1\times
{\mathcal Z}^2$ and study it by
  Hirota's bilinear method. This equation has a singular continuum limit into a
system which admits the reduction to the Davey-Stewartson equation. The
solutions of this discrete DS system are characterized by Casorati and Grammian
determinants. Based on the bilinear form of this discrete DS system, we
construct the bilinear B\"{a}cklund transformation which enables us to obtain
its Lax pair.
