We study the growth and saturation of the superradiant instability of a complex, massive vector (Proca) field as it extracts energy and angular momentum from a spinning black hole, using numerical solutions of the full Einstein-Proca equations. We concentrate on a rapidly spinning black hole (a = 0.99) and the dominant m = 1 azimuthal mode of the Proca field, with real and imaginary components of the field chosen to yield an axisymmetric stress-tensor, hence spacetime. We find that in excess of 9% of the black hole's mass can be transferred into the field. In all cases studied, the superradiant instability smoothly saturates when the black hole's horizon frequency decreases to match the frequency of the Proca cloud that spontaneously forms around the black hole.Introduction.-A remarkable feature of spinning black holes (BHs) is that a portion of their massup to 29% for extremal spin -can in principle be extracted. One way to realize this liberation of rotational energy is through the interaction of the BH with an impinging wave -be it scalar, electromagnetic, or gravitational -with frequency ω < mΩ BH , where Ω BH is the BH horizon frequency, and m is the azimuthal number of the wave. Waves satisfying this criterion exhibit superradiance, and carry away energy and angular momentum from the BH. An analogous phenomenon can occur for charged BHs, where the electromagnetic energy of the BH is superradiantly transferred to an interacting charged matter field interacting with the BH.Going back to [1] there has been speculation of how superradiance could be combined with a confining mechanism to force the wave to continuously interact with the BH and hence undergo exponential growth -a so called "black hole bomb." The first nonlinear studies of this process were recently undertaken for a charged scalar field around a charged BH in spherical symmetry, both in a reflective cavity in asymptotically flat space [2], and in the naturally confining environment of an asymptotically Anti-de Sitter domain [3].However, there is an exciting possibility that a variation of this scenario could in fact be realized around astrophysical spinning BHs. Massive bosonic fields with Compton wavelength comparable to, or larger than, the horizon radius of a BH can form bound states around the BH, and if the latter is spinning the bound states can grow from a seed perturbation through superradiance [4][5][6]. This implies that stellar mass BHs can probe the existence of ultralight bosons with masses 10 −10 eV that are weakly coupled to ordinary matter and thus difficult to detect by other