A new path-following primal-dual augmented Lagrangian method is proposed for solving nonlinear equality constrained optimization problems (NEP). At each iteration, a Newton-like method is used to solve a perturbed optimality condition that defines a penalty trajectory parameterized by both the penalty parameter and the estimated Lagrange multipliers. We show that this method is globally convergent and has a quadratic convergence rate in the limit. Finally, numerical experiments on problems from the CUTEst test collection are are used to support the theoretical analysis.