  We show that monopoles can be pair produced by cosmological magnetic fields
in the early universe. The pair production gives rise to relic monopoles, and
at the same time induces a self-screening of the magnetic fields. By studying
these effects we derive limits on the monopole mass, and also on the initial
amplitude of primordial magnetic fields. Monopoles of GUT scale mass can even
be produced if primordial magnetic fields exist at sufficiently high redshifts.
