Au/Ag core/shell nanorods with different shell thickness were synthesized in aqueous solution by chemically depositing silver on gold nanorods surface. With the silver coating, the longitudinal plasmon mode of the nanorods shifted blue and was enhanced. A dipole-limit calculation, based on confocal ellipsoids, simulates the spectra of the core/shell nanorods using bulk dielectric functions. Good agreement with the experimental result was achieved. Light scattering spectra of single nanorods were taken by dark-field microscopy to measure the homogeneous line width. The scattering spectra of single gold nanorods are less than 10% broader than the theoretical value, while the spectra of silver-coated nanorods are systematically 40-50% broader. The additional damping of the plasmon was modeled as the extra scattering at the Au-Ag interface and the nanorods surface. A model for evaluating the plasmon damping in inhomogeneous metallic systems with interfaces is presented.