The problem of ensuring that a control input remains within certain bounds is ubiquitous in all control applications. This problem becomes even more difficult in the context of adaptive control, since the plant to be controlled has unknown parameters. In this paper we establish conditions under which minimum phase discrete-time plants can be adaptively controlled in a stable manner in the presence of amplitude constraints on the input. All open-loop stable plants are shown to result in global stability with the proposed adaptive controller, and open-loop unstable plants are shown to result in local stability when the initial conditions of the closed-loop system lie within a compact set. It is also shown that when initial conditions exceed a certain value, open-loop unstable plants can become unstable. It is furthermore shown that the adaptive algorithms can be appropriately modified in the presence of exogeneous disturbances and certain modeling errors to yield boundedness. Simulations are presented to support the theoretical derivations.