We demonstrate the existence of a new steady-state magnetic reconnection configuration which lies at the boundary of the basins of attraction between the Sweet-Parker and Hall reconnection configurations. The solution is linearly unstable to small perturbations and its identification required a novel iterative numerical technique. The eigenmodes of the unstable solution are localized near the X line, suggesting that the onset of fast reconnection in a weakly collisional plasma is initiated locally at the X line as opposed to remotely at the boundaries.