We extend the Parker bound on the galactic flux F of magnetic monopoles. By
requiring that a small initial seed held must survive the collapse of the
protogalaxy, before any regenerative dynamo effects become significant, we
develop a stranger bound. The survival and continued growth of an initial g
alactic seed field less than or equal to 10(-9) G demand that F less than o
r equal to 5 x 10(-21) (m/10(17) GeV)cm(-2) sec(-1) sr(-1). For a given mon
opole mass, this bound is four and a half orders of magnitude more stringen
t than the previous "extended Parker bound," but is more speculative as it
depends on assumptions about the behavior of magnetic fields during protoga
lactic collapse. For monopoles which do not overclose the universe (Omega(m
)<l), the maximum flux allowed is now 8 x 10(-19) cm(-2) s(-1) sr(-1), a fa
ctor of 150 lower than the maximum flux allowed by the extended Parker boun
d.