In this paper, we consider the wave equation with Dirichlet boundary control subject to a nonlinearity, the kind of which includes (but is not restricted to) pointwise saturation mappings. The case where only a subset of the boundary is actuated is allowed. Initial data is taken in the optimal energy space associated with Dirichlet boundary control-which means that we deal with (very) weak solutions. Using nonlinear semigroup techniques, we prove that the associated closed-loop system is asymptotically stable. Some numerical simulations are given to illustrate the stability result.