The nonlinear Enskog equation with a discretized spatial variable is studied in a Banach space of absolutely integrable functions of the velocity variables. The Enskog equation is a kinetic equation of Boltzmann typc which, unlike the Boltzmann equation, is applicable to gases in the moderately dense regime. In this lattice model the generator of free streaming is replaced by a finite difference operator. Conservation laws and positivity are utilized to extend local solutions of a cutoff model to global solutions. Then compactness arguments lead to the existence of weak global solutions of the Enskog lattice equation. Molecular interactions are introduced via a next-nearest neighbor potential, thereby modeling a square well potential.