We propose a nonlinear extension of the Fierz-Pauli mass for the graviton through a functional of the vielbein and an external Minkowski background. The functional generalizes the notion of the measure, since it reduces to a cosmological constant if the external background is formally sent to zero. Such a term and the explicit external background emerge dynamically from a bi-gravity theory, having both a massless and a massive graviton in its spectrum, in a specific limit in which the massless mode decouples, while the massive one couples universally to matter. We investigate the massive theory using the Stückelberg method and providing a 't Hooft-Feynman gauge fixing, in which the tensor, vector and scalar Stückelberg fields decouple. We show that this model has the softest possible ultraviolet behavior that can be expected from any generic (Lorentz-invariant) theory of massive gravity, namely that it becomes strong only at the scale Λ3=(mg 2MP)1/3.