  We report an investigation of the structural and electronic properties of
hybrid superconductor/ferromagnet (S/F) bilayers of composition
Nb/Cu$_{60}$Ni$_{40}$ prepared by magnetron sputtering. X-ray and neutron
reflectometry show that both the overall interfacial roughness and vertical
correlations of the roughness of different interfaces are lower for
heterostructures deposited on Al$_2$O$_3$(1$\bar{1}$02) substrates than for
those deposited on Si(111). Mutual inductance experiments were then used to
study the influence of the interfacial roughness on the superconducting
transition temperature, $T_C$. These measurements revealed a $\sim$ 4% higher
$T_C$ in heterostructures deposited on Al$_2$O$_3$, compared to those on Si. We
attribute this effect to a higher mean-free path of electrons in the S layer,
caused by a suppression of diffusive scattering at the interfaces. However, the
dependence of the $T_C$ on the thickness of the ferromagnetic layer is not
significantly different in the two systems, indicating a weak influence of the
interfacial roughness on the transparency for Cooper pairs.
