  The broken inversion symmetry at the surface of a metallic film (or, more
generally, at the interface between a metallic film and a different metallic or
insulating material) greatly amplifies the influence of the spin-orbit
interaction on the surface properties. The best known manifestation of this
effect is the momentum-dependent splitting of the surface state energies
(Rashba effect). Here we show that the same interaction also generates a
spin-polarization of the bulk states when an electric current is driven through
the bulk of the film. For a semi-infinite jellium model, which is
representative of metals with a closed Fermi surface, we prove as a theorem
that, regardless of the shape of the confinement potential, the induced surface
spin density at each surface is given by ${\bf S} =-\gamma \hbar {\bf \hat
z}\times {\bf j}$, where ${\bf j}$ is the particle current density in the bulk,
${\bf \hat z}$ the unit vector normal to the surface, and
$\gamma=\frac{\hbar}{4mc^2}$ contains only fundamental constants. For a general
metallic solid $\gamma$ becomes a material-specific parameter that controls the
strength of the interfacial spin-orbit coupling. Our theorem, combined with an
{\it ab initio} calculation of the spin polarization of the current-carrying
film, enables a determination of $\gamma$, which should be useful in modeling
the spin-dependent scattering of quasiparticles at the interface.
