The interaction between plasmonic and excitonic systems and the formation of hybridized states is an area of intense interest, due to the potential to create exotic light-matter states. We report herein coupling between the leaky surface plasmon polariton (SPP) modes of single Ag nanowires and excitons of a cyanine dye (TDBC) in an open nanocavity. Silver nanowires were spin-cast onto glass coverslips, and the wavevector of the leaky SPP mode was measured by back focal plane (BFP) microscopy. Performing these measurements at different wavelengths allows the generation of dispersion curves, which show avoided crossings after deposition of a concentrated TDBC-PVA film. The Rabi splitting frequencies (Ω) determined from the dispersion curves vary between nanowires, with a maximum value of Ω = 390 ± 80 meV. The experiments also show an increase in attenuation of the SPP mode in the avoided crossing region. The ability to measure attenuation for the hybrid exciton-SPP states is a powerful aspect of these single nanowire experiments, as this quantity is not readily available from ensemble experiments.