Context : Aims : A close companion of \zeta Orionis A was found in 2000 with the Navy Precision Optical Interferometer ( NPOI ) , and shown to be a physical companion . Because the primary is a supergiant of type O , for which dynamical mass measurements are very rare , the companion was observed with NPOI over the full 7-year orbit . Our aim was to determine the dynamical mass of a supergiant that , due to the physical separation of more than 10 AU between the components , can not have undergone mass exchange with the companion . Methods : The interferometric observations allow measuring the relative positions of the binary components and their relative brightness . The data collected over the full orbital period allows all seven orbital elements to be determined . In addition to the interferometric observations , we have analyzed archival spectra obtained at the Calar Alto , Haute Provence , Cerro Armazones , and La Silla observatories , as well as new spectra obtained at the VLT on Cerro Paranal . In the high-resolution spectra we identified a few lines that can be associated exclusively to one or the other component for the measurement of the radial velocities of both . The combination of astrometry and spectroscopy then yields the stellar masses and the distance to the binary star . Results : The resulting masses for components Aa of 14.0 \pm 2.2 M _ { \odot } and Ab of 7.4 \pm 1.1 M _ { \odot } are low compared to theoretical expectations , with a distance of 294 \pm 21 pc which is smaller than a photometric distance estimate of 387 \pm 54 pc based on the spectral type B0III of the B component . If the latter ( because it is also consistent with the distance to the Orion OB1 association ) is adopted , the mass of the secondary component Ab of 14 \pm 3 M _ { \odot } would agree with classifying a star of type B0.5IV . It is fainter than the primary by about 2.2 \pm 0.1 magnitudes in the visual . The primary mass is then determined to be 33 \pm 10 M _ { \odot } . The possible reasons for the distance discrepancy are most likely related to physical effects , such as small systematic errors in the radial velocities due to stellar winds . Conclusions :