We construct a fully covariant theory of massive gravity which does not require the introduction of an external reference metric, and overcomes the usual problems of massive gravity theories (fatal ghosts instabilities, acausality and/or van Dam-Veltman-Zakharov discontinuity). The equations of motion of the theory are nonlocal but respect causality. The starting point is the quadratic action proposed in the context of the degravitation idea. We show that it is possible to extend it to a fully nonlinear covariant theory. This theory describes the 5 degrees of freedom of a massive graviton plus a scalar ghost. However, contrary to generic nonlinear extensions of Fierz-Pauli massive gravity, the ghost has the same mass m as the massive graviton, independently of the background, and smoothly goes into a nonradiative degree of freedom for m→0. As a consequence, for m∼H0 the vacuum instability induced by the ghost is irrelevant even over cosmological time scales. We finally show that an extension of the model degravitates a vacuum energy density of order M4Pl down to a value of order M2Plm2, which for m=O(H0) is of order of the observed value of the vacuum energy density.