We design a full-order observer for discrete-time linear time-invariant systems with constant output delays. The observer design is based on the output delay model expressed by a two-dimensional state variable, with discrete-time and space independent variables. Employing a discrete-time state transformation, we construct an explicit strict Lyapunov function that enables us to prove the global exponential stability of the full-order observer error system with an explicit estimate of the exponential decay rate. The numerical example demonstrates the design of the full-order observer and illustrates the validity of the exponential stability.