Silver nanostructures with narrow plasmon linewidths and good chemical stability are strongly desired for plasmonic applications. Herein, a facile method is discussed for the preparation of Ag nanostructures with narrow plasmon linewidths and improved chemical stability through Ag overgrowth on monodispersed Au nanobipyramids. Structural evolution from bipyramid through rice to rod is observed, indicating that Ag atoms are preferentially deposited on the side surfaces of Au nanobipyramids. The resultant (Au nanobipyramid)@Ag nanostructures possess high size and shape uniformities, and much narrower plasmon linewidths than other Ag nanostructures. The spectral evolution of the supported Ag nanostructures is ascertained by both ensemble and single-particle characterizations, together with electrodynamic simulations. Systematic measurements of the refractive index sensing characteristics indicate that Ag nanostructures in this study possess high index sensitivities and figure of merit (sensitivity divided by linewidth) values. Moreover, Ag nanostructures in this study exhibit greatly improved chemical stability. The superior sensing capability of Ag nanostructures in this study is further demonstrated by the detection of sulfide ions at a relatively low detection limit. Taken together, results of this study show that the Au-nanobipyramid-supported Ag nanostructures will be an outstanding candidate for the design of ultrasensitive plasmonic sensing devices as well as for the development of other plasmon-enabled technological applications.