We prove a local index theorem of Atiyah-Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). After introducing a renormalized supertrace on Lie manifolds with spin structure, defined on a suitable class of rapidly decaying functions, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of the rescaled bundle over the adiabatic groupoid and introduce a rescaling of the heat kernel encoded in a vector bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion using the Lichnerowicz theorem on the fibers of the groupoid and the Lie manifold.