An optimal control problem for an elliptic obstacle variational inequality is considered. The obstacle is taken to be the control and the solution to the obstacle problem is taken to be the state, The goal is to find the optimal obstacle from H-0(1)(Omega) so that the state is dose to the desired profile while the H-1(Omega) norm of the obstacle is not too large, Existence, uniqueness, and regularity as well as some characterizations of the optimal pairs are established.