While conducting loops are in principle sensitive to external magnetic field, as is pronouncedly exemplifies by the Aharonov-Bohm (AB) effect, the small radius of molecular rings makes the observation of such effects challenging. Indeed, the unrealistically large magnetic field needed to realize the AB effect in molecular rings has led to a widespread belief that molecular conduction is insensitive to laboratory realizable fields. Here we revisit this issue, presenting conditions under which magnetic field control of molecular ring conduction is realizable with pronounced effects on the I-V characteristics. We find these conditions to be (a) weak molecule-leads coupling, implying relatively distinct conduction resonances, (b) asymmetric junction structure (e.g. meta- or ortho- connected benzene rather than a para structure) and (c) minimal dephasing (implying low temperature) so as to maintain coherence between multiple pathways of conduction.