Recent nonlinear completions of Fierz-Pauli theory for a massive spin-2 field include nonlinear massive gravity and bimetric theories. The spectrum of black-hole solutions in these theories is rich and comprises the same vacuum solutions of Einstein's gravity enlarged to include a cosmological constant. It was recently shown that Schwarzschild (de Sitter) black holes in these theories are generically unstable against spherical perturbations. Here we show that a notable exception is partially massless gravity, where the mass of the graviton is fixed in terms of the cosmological constant by mu(2) = 2 Lambda/3 and a new gauge invariance emerges. We find that general relativity black holes are stable in this limit. Remarkably, the spectrum of massive gravitational perturbations is isospectral.