  We study the effect of interface disorder on the spin-dependent interface
resistances of Co/Cu, Fe/Cr and Au/Ag multilayers using a newly developed
method for calculating transmission matrices from first-principles. The
efficient implementation using tight-binding linear-muffin-tin orbitals allows
us to model interface disorder using large lateral supercells whereby specular
and diffuse scattering are treated on an equal footing. Without introducing any
free parameters, quantitative agreement with experiment is obtained. We predict
that disorder {\it reduces} the majority-spin interface resistance of
Fe/Cr(100) multilayers by a factor 3.
