The resonant interaction between quasi-one-dimensional excitons and photons is investigated. For a single isolated organic quantum wire, embedded in its single crystal monomer matrix, the strong exciton-photon coupling regime is reached without the need of a cavity for photons. This is evidenced by the suppression of the resonant excitonic absorption arising when the system eigenstate is a polariton. The resonant excitonic absorption can be understood in terms of a balance between the exciton coherence time and the Rabi period between excitonlike and photonlike states of the polariton.