The problem of relaxation of optimal control problems with state and control constraints is formulated in this paper. We determine that if the original problem consists of minimizing, over control functions zeta(.), g(xi(T)) subject to d xi/ds = f(s, xi, zeta), t < s less than or equal to T, and h(s,xi(s), zeta(s)) less than or equal to 0 for a.e. t less than or equal to s less than or equal to T, then the relaxed problem consists of minimizing, over measure-valued control functions mu(.), g(<(xi)over cap (T))>, subject to d xi/ds = integral(z), f(s, xi(s), z)mu(s, dz) and mu(s) - ess sup(z) h(s, <(xi)over cap (s)>,z) less than or equal to 0 for a.e, t less than or equal to s less than or equal to T. For each s this is the essential supremum of h in z with respect to the measure mu(s).