The magnetic-field influence on the inelastic interelectrode tunnel current mediated by a molecular wire is studied theoretically for the case in which the wire includes paramagnetic ions with frozen angular momenta. The vibrational and spin relaxation within the terminal sites of the wire (adjacent to the electrodes) are assumed to be much faster than the related jump of the tunneling electron. For a wire containing a pair of paramagnetic ions, a low-temperature blocking of the current is demonstrated for the cases of ions both coupled by an exchange interaction and largely separated in space. The blocking is shown to appear if the paramagnetic ion reduces its electronic ground-state spin S to S-1/2 during the formation of the intermediate bound state with the transferred electron. At low temperatures the current formation is predominantly originated thus by tunnel channels for which the magnetic energy of paramagnetic ions has a minimal value. In the presence of an exchange spin-spin interaction or a single-ion anisotropy, just these predominant channels are manifested to form a steplike behavior of the current before the current itself is completely blocked by the increasing magnetic field.