  For a sequence $\{\alpha_n\}_{n=0}^\infty$, we consider the Hankel operator
$\Gamma_\alpha$, realised as the infinite matrix in $\ell^2$ with the entries
$\alpha_{n+m}$. We consider the subclass of such Hankel operators defined by
the "double positivity" condition $\Gamma_\alpha\geq0$,
$\Gamma_{S^*\alpha}\geq0$; here $S^*\alpha$ is the shifted sequence
$\{\alpha_{n+1}\}_{n=0}^\infty$. We prove that in this class, the sequence
$\alpha$ is uniquely determined by the spectral shift function $\xi_\alpha$ for
the pair $\Gamma_\alpha^2$, $\Gamma_{S^*\alpha}^2$. We also describe the class
of all functions $\xi_\alpha$ arising in this way and prove that the map
$\alpha\mapsto\xi_\alpha$ is a homeomorphism in appropriate topologies.
