We report on the spectroscopic confirmation of a large-scale structure around the luminous z=6.31 quasi-stellar object (QSO) SDSS J1030+0524, powered by a one billion solar mass black hole. The structure is populated by at least six members, namely, four Lyman-break galaxies (LBGs), and two Lyman alpha emitters (LAEs). The four LBGs were identified among a sample of 21 i-band dropouts with z(AB) < 25.5 selected up to projected separations of 5 physical Mpc (15 arcmin) from the QSO. Their redshifts were determined through multi-object spectroscopic observations at 8-10 m class telescopes lasting up to eight hours. The two LAEs were identified in a 6 h VLT/MUSE observation centered on the QSO. The redshifts of the six galaxies cover the range between 6.129-6.355. Assuming that the peculiar velocities are negligible, this range corresponds to radial separations of +/- 5 physical Mpc from the QSO, that is comparable to the projected scale of the observed LBG distribution on the sky. We conservatively estimate that this structure is significant at a level > 3.5 sigma and that the level of the galaxy overdensity is at least 1.5-2 within the large volume sampled (similar to 780 physical Mpc(3)). The spectral properties of the six member galaxies (Ly alpha strength and UV luminosity) are similar to those of field galaxies at similar redshifts. This is the first spectroscopic identification of a galaxy overdensity around a supermassive black hole in the first billion years of the Universe. Our finding lends support to the idea that the most distant and massive black holes form and grow within massive (> 10(12) M-circle dot) dark matter halos in large-scale structures and that the absence of earlier detections of such systems is likely due to observational limitations.