  We develop a Non-Crossing Approximation (NCA) for the effective cluster
problem of the recently developed Dynamical Cluster Approximation (DCA). The
DCA technique includes short-ranged correlations by mapping the lattice problem
onto a self-consistently embedded periodic cluster of size $N_c$. It is a fully
causal and systematic approximation to the full lattice problem, with
corrections ${\cal{O}}(1/N_c)$ in two dimensions. The NCA we develop is a
systematic approximation with corrections ${\cal{O}}(1/N_c^3)$. The method will
be discussed in detail and results for the one-particle properties of the
Hubbard model are shown. Near half filling, the spectra display pronounced
features including a pseudogap and non-Fermi-liquid behavior due to
short-ranged antiferromagnetic correlations.
