MWC 656 (=HD 215227) was recently discovered to be the first binary system composed of a Be star and a black hole (BH). We observed it with XMM-Newton, and detected a faint X-ray source compatible with the position of the optical star, thus proving it to be the first Be/BH X-ray binary. The spectrum analysis requires a model fit with two components, a blackbody plus a power law, with k_BT = 0.07^{+0.04}_{-0.03} keV and a photon index Γ = 1.0 ± 0.8, respectively. The non-thermal component dominates above sime0.8 keV. The obtained total flux is F(0.3-5.5\, keV) = (4.6^{+1.3}_{-1.1})\times 10^{-14} erg cm–2 s–1. At a distance of 2.6 ± 0.6 kpc the total flux translates into a luminosity L X = (3.7 ± 1.7) × 1031 erg s–1. Considering the estimated range of BH masses to be 3.8-6.9 M ☉, this luminosity represents (6.7 ± 4.4) × 10–8 L Edd, which is typical of stellar-mass BHs in quiescence. We discuss the origin of the two spectral components: the thermal component is associated with the hot wind of the Be star, whereas the power-law component is associated with emission from the vicinity of the BH. We also find that the position of MWC 656 in the radio versus X-ray luminosity diagram may be consistent with the radio/X-ray correlation observed in BH low-mass X-ray binaries. This suggests that this correlation might also be valid for BH high-mass X-ray binaries (HMXBs) with X-ray luminosities down to ~10–8 L Edd. MWC 656 will allow the accretion processes and the accretion/ejection coupling at very low luminosities for BH HMXBs to be studied.