  We study Bose-Einstein condensation (BEC) in one-dimensional noninteracting
Bose gases in Poisson random potentials on $\mathbb R$ with single-site
potentials that are nonnegative, compactly supported, and bounded measurable
functions in the grand-canonical ensemble at positive temperatures in the
thermodynamic limit. For particle densities that are larger than a critical
one, we prove the following: With arbitrarily high probability when choosing
the fixed strength of the random potential sufficiently large, BEC where only
the ground state is macroscopically occupied occurs. If the strength of the
Poisson random potential converges to infinity in a certain sense but
arbitrarily slowly, then this kind of BEC occurs in probability and in the
$r$th mean, $r \ge 1$. Furthermore, in Poisson random potentials of any fixed
strength an arbitrarily high probability for type-I g-BEC is also obtained by
allowing sufficiently many one-particle states to be macroscopically occupied.
