  A pseudo-cone in ${\mathbb R}^n$ is a nonempty closed convex set $K$ not
containing the origin and such that $\lambda K \subseteq K$ for all $\lambda\ge
1$. It is called a $C$-pseudo-cone if $C$ is its recession cone, where $C$ is a
pointed closed convex cone with interior points. The cone-volume measure of a
pseudo-cone can be defined similarly as for convex bodies, but it may be
infinite. After proving a necessary condition for cone-volume measures of
$C$-pseudo-cones, we introduce suitable weights for cone-volume measures,
yielding finite measures. Then we provide a necessary and sufficient condition
for a Borel measure on the unit sphere to be the weighted cone-volume measure
of some $C$-pseudo-cone.
