  We extend the Parker bound on the galactic flux $\cal F$ of magnetic
monopoles. By requiring that a small initial seed field must survive the
collapse of the protogalaxy, before any regenerative dynamo effects become
significant, we develop a stronger bound. The survival and continued growth of
an initial galactic seed field $\leq 10^{-9}$G demand that ${\cal F} \leq 5
\times 10^{-21} (m/10^{17} {GeV}) {cm}^{-2} {sec}^{-1} {sr}^{-1}$. For a given
monopole mass, this bound is four and a half orders of magnitude more stringent
than the previous `extended Parker bound', but is more speculative as it
depends on assumptions about the behavior of magnetic fields during
protogalactic collapse. For monopoles which do not overclose the Universe
($\Omega_m <1$), the maximum flux allowed is now $8 \times 10^{-19}$ cm^{-2}
s^{-1} sr^{-1}, a factor of 150 lower than the maximum flux allowed by the
extended Parker bound.
