Let $\psi$ be a given function defined on a Riemannian space. Under
what conditions does there exist a compact starshaped hypersurface $M$
for which $\psi$, when evaluated on $M$, coincides with the $m$-th
elementary symmetric function of principal curvatures of $M$ for a given
$m$? The corresponding existence and uniqueness problems in Euclidean
space have been investigated by several authors in the mid 1980s.
Recently, conditions for existence were established in elliptic space
and, most recently, for hyperbolic space. However, the uniqueness
problem has remained open. In this paper we investigate the problem of
uniqueness in hyperbolic space and show that uniqueness (up to a
geometrically trivial transformation) holds under the same conditions
under which existence was established.