  The chemical potential of a hard-sphere fluid can be expressed in terms of
the contact value of the radial distribution function of a solute particle with
a diameter varying from zero to that of the solvent particles. Exploiting the
explicit knowledge of such a contact value within the Percus--Yevick (PY)
theory, and using standard thermodynamic relations, a hitherto unknown PY
equation of state, $p/\rho k_BT=-(9/\eta)\ln(1-\eta)-(16-31\eta)/2(1-\eta)^2$,
is unveiled. This equation of state turns out to be better than the one
obtained from the conventional virial route. Interpolations between the
chemical-potential and compressibility routes are shown to be more accurate
than the widely used Carnahan--Starling equation of state. The extension to
polydisperse hard-sphere systems is also presented.
