In this article we give a new technique for exponentiating infinite dimensional graded representations of graded Lie algebras that allows for the exponentiation of some non-locally nilpotent elements. Our technique is to naturally extend the representation of the Lie algebra g on the spaceVnaturally to a representation on a subspace £ of the dual spaceV*. After introducing the technique, we prove that it enables the exponentiation of all elements of free Lie Algebras and afhne Kac-Moody Lie algebras.