The Runge-Kutta discretization of optimal control problems governed by ODEs is considered and the convergence rate is determined. The analysis uses a connection between the Kuhn-Tucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle. This connection can also be exploited in numerical solution techniques that require the gradient of the discrete cost function. The transformation of the first-order necessary conditions for the discrete control problem leads to an RK scheme for the adjoint equation which can be different from the original RK discretization of the state equation. The coefficients of the RK scheme must therefore satisfy some additional conditions in order to obtain third or fourth-order accuracy for the control problem.