The DPG method with optimal test functions for solving linear quadratic optimal control problems with control constraints is studied. We prove existence of a unique optimal solution of the nonlinear discrete problem and characterize it through first -order optimality conditions. Furthermore, we systematically develop a priori as well as a posteriori error estimates. Our proposed method can be applied to a wide range of constrained optimal control problems subject to, e.g., scalar second -order PDEs and the Stokes equations. Numerical experiments that illustrate our theoretical findings are presented.