We have recently discovered a supermassive binary black hole system with a projected separation between the two black holes of 7.3 pc in the radio galaxy 0402+379 (Rodriguez et al. 2006). This is the most compact supermassive binary black hole pair yet imaged by more than two orders of magnitude. We present Global VLBI observations at 1.3464 GHz of this radio galaxy, taken to improve the quality of the H I data. Two absorption lines are found toward the southern jet of the source, one redshifted by 370 +/- 10 km s(-1) and the other blueshifted by 700 +/- 10 km s(-1) with respect to the systemic velocity of the source, which, along with the results obtained for the opacity distribution over the source, suggests the presence of two mass clumps rotating around the central region of the source. We propose a model consisting of a geometrically thick disk, of which we only see a couple of clumps, that reproduces the velocities measured from the Hi absorption profiles. These clumps rotate in circular Keplerian orbits around an axis that crosses one of the supermassive black holes of the binary system in 0402+379. We find an upper limit for the inclination angle of the twin jets of the source to the line of sight of theta = 66 degrees, which, according to the proposed model, implies a lower limit on the central mass of similar to 7 x 10(8) M(circle dot) and a lower limit for the scale height of the thick disk of similar to 12 pc.