  For a finite, positive, Borel measure $\mu$ on $(0,1)$ we consider an
infinite matrix $\Gamma_\mu$, related to the classical Hausdorff matrix defined
by the same measure $\mu$, in the same algebraic way that the Hilbert matrix is
related to the Ces\'aro matrix. When $\mu$ is the Lebesgue measure,
$\Gamma_\mu$ reduces to the classical Hilbert matrix. We prove that the
matrices $\Gamma_\mu$ are not Hankel, unless $\mu$ is a constant multiple of
the Lebesgue measure,
  we give necessary and sufficient conditions for their boundedness on the
scale of Hardy spaces $H^p, \, 1 \leq p < \infty$, and we study their
compactness and complete continuity properties. In the case $2\leq p<\infty$,
we are able to compute the exact value of the norm of the operator.
