  We study a spatial branching model, where the underlying motion is
$d$-dimensional ($d\ge1$) Brownian motion and the branching rate is affected by
a random collection of reproduction suppressing sets dubbed mild obstacles. The
main result of this paper is the quenched law of large numbers for the
population for all $d\ge1$. We also show that the branching Brownian motion
with mild obstacles spreads less quickly than ordinary branching Brownian
motion by giving an upper estimate on its speed. When the underlying motion is
an arbitrary diffusion process, we obtain a dichotomy for the quenched local
growth that is independent of the Poissonian intensity. More general offspring
distributions (beyond the dyadic one considered in the main theorems) as well
as mild obstacle models for superprocesses are also discussed.
