We report on the strong coupling of surface plasmon polaritons and molecular vibrations in an organic-inorganic plasmonic hybrid structure consisting of a ketone-based polymer deposited on top of a silver layer. Attenuated-total-reflection spectra of the hybrid reveal an anticrossing in the dispersion relation in the vicinity of the carbonyl stretch vibration of the polymer with an energy splitting of the upper and lower polariton branch up to 15 meV. The splitting is found to depend on the molecular layer thickness and saturates for micrometer-thick films. This new hybrid state holds a strong potential for application in chemistry and optoelectronics.