  A nonempty closed convex set in ${\mathbb R}^n$, not containing the origin,
is called a pseudo-cone if with every $x$ it also contains $\lambda x$ for
$x\ge 1$. We consider pseudo-cones with a given recession cone $C$, called
$C$-pseudo-cones. The family of $C$-pseudo-cones can, with reasonable
justification, be considered as a counterpart to the family of convex bodies
containing the origin in the interior. For a $C$-pseudo-cone one can naturally
define a surface area measure and a covolume. Since they are in general
infinite, we introduce a weighting, leading to modified versions of surface
area and covolume. These are finite and still homogeneous, though of different
degrees. Our main result is a Minkowski type existence theorem for
$C$-pseudo-cones with given weighted surface area measure.
