This chapter considers a linear-quadratic optimal control problem, where the control variable enters the state equation via the Dirichlet boundary conditions. We refer to this type of problem, as Dirichlet boundary control problem. The intrinsic difficulty is that the Dirichlet boundary conditions do not directly enter the classical variational formulation of an elliptic equation. Thus, we base our problem setting on a very weak formulation of the state equation. We derive optimality conditions and provide regularity results, which especially show that for the optimal solutions, a classical weak formulation can be used. The main results of the chapter are a priori error estimates for the discretization of this class of problems.