We present a new formulation of effective field theory for nucleon-nucleon (NN) interactions which treats pion interactions perturbatively, and we offer evidence that the expansion converges satisfactorily to third order in the expansion, which we have computed analytically for s and d wave NN scattering. Starting with the Kaplan-Savage-Wise (KSW) expansion about the nontrivial fixed point corresponding to infinite NN scattering length, we cure the convergence problems with that theory by summing to all orders the singular short distance part of the pion tensor interaction. This method makes possible a host of high precision analytic few-body calculations in nuclear physics.