<p>A pseudo-cone in ℝ^<i>n</i> is a nonempty closed convex set <i>K</i> not containing the origin and such that <i>λK</i> ⊆ <i>K</i> for all <i>λ</i> ≥ 1. It is called a <i>C</i>-pseudo-cone if <i>C</i> is its recession cone, where <i>C</i> 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 <i>C</i>-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 <i>C</i>-pseudo-cone.</p>