  Let $D$ be an integral domain and $\Gamma$ be a torsion-free commutative
cancellative (additive) semigroup with identity element and quotient group $G$.
In this paper, we show that if char$(D)=0$ (resp., char$(D)=p>0$), then
$D[\Gamma]$ is a weakly Krull domain if and only if $D$ is a weakly Krull
UMT-domain, $\Gamma$ is a weakly Krull UMT-monoid, and $G$ is of type $(0,0,0,
\dots )$ (resp., type $(0,0,0, \dots )$ except $p$). Moreover, we give
arithmetical applications of this result.
