Let $H$ be a Krull monoid with infinite cyclic class group $G$ and let $G_P
\subset G$ denote the set of classes containing prime divisors. We study under
which conditions on $G_P$ some of the main finiteness properties of
factorization theory--such as local tameness, the finiteness and rationality of
the elasticity, the structure theorem for sets of lengths, the finiteness of
the catenary degree, and the existence of monotone and of near monotone chains
of factorizations--hold in $H$. In many cases, we derive explicit
characterizations.