We compare two constraints on the strength of the cosmological primord
ial magnetic field: the one following from the restrictions on the Dir
ac neutrino spin flip in the early Universe, and another one based on
the galactic dynamo theory for the Milky Way (presuming that the seed
magnetic field has a relic origin). Since the magnetic field facilitat
es transitions between left- and right-handed neutrino states, thereby
affecting He-4 production at primordial nucleosynthesis, we can obtai
n a guaranteed upper limit on the strength of the relic magnetic field
in the protogalaxy, B(c) less than or similar to 4 X 10(-9)-3 X 10(13
) G, depending on the neutrino magnetic moment, if we adopt the MSW ex
planation of the GALLEX results. On the other hand, models of the dyna
mo in the Milky Way indicate that the seed magnetic field should be at
least 10(-11)-10(-13) G at the protogalaxy scale L = 100 kpc. These u
pper and lower limiting ranges are marginally consistent provided the
electron neutrino mass is below 0.3 eV. The results apply to a relic m
agnetic field produced in the early Universe by any causal mechanism b
efore the nucleosynthesis.