We consider a nonautonomous problem of the calculus of variations where the Lagrangian is Borel measurable; we allow state, velocity and endpoint constraints. We prove that if the Lagrangian satisfies a nonsmooth version of Cesari's Condition (S), then any W^{1,1} local minimizer is Lipschitz whenever Λ is coercive. The proof is obtained via a new variational inequality, called also directional Weierstrass type condition, formulated here, that holds under the extended Condition (S) (just Borel measurability if Λ is autonomous). The proof of the directional Weierstrass type condition (W) is based on Clarke's nonsmooth recent versions of the Maximum Principle.