  For a large class (heuristically most) of irreducible binary cubic forms
$F(x,y) \in \mathbb Z[x,y]$, Bennett and Dahmen proved that the generalized
superelliptic equation $F(x,y)=z^l$ has at most finitely many solutions in $x,y
\in \mathbb Z$ coprime, $z \in \mathbb Z$ and exponent $l \in \mathbb Z_{\geq
4} $. Their proof uses, among other ingredients, modularity of certain mod $l$
Galois representations and Ribet's level lowering theorem. The aim of this
paper is to treat the same problem for binary cubics with coefficients in
$\mathcal O_K$, the ring of integers of an arbitrary number field $K$, using by
now well-documented modularity conjectures.
