Suppose that
P
P
is a convex polyhedron in the hyperbolic
3
3
-space with finite volume and
P
P
has integer
(
>
1
)
( > 1)
submultiples of
π
\pi
as dihedral angles. We prove that if the rank of the abelianization of a normal torsion-free finite index subgroup of the polyhedral group
G
G
associated to
P
P
is one, then
P
P
has exactly one ideal vertex of type
(
2
,
2
,
2
,
2
)
(2,2,2,2)
and
G
G
has an index two subgroup which does not contain any one of the four standard generators of the stabilizer of the ideal vertex.