The modification and control of exciton-photon interactions in semiconductors is of both fundamental and practical interest, being of direct relevance to the design of improved light-emitting diodes, photodetectors and lasers. In a semiconductor microcavity, the confined electromagnetic field modifies the optical transitions of the material. Two distinct types of interaction are possible: Weak and strong coupling. In the former perturbative regime, the spectral and spatial distribution of the emission is modified but exciton dynamics are little altered. In the latter case, however, mixing of exciton and photon states occurs leading to strongly modified dynamics. Both types of effect have been observed in planar microcavity structures in inorganic semiconductor quantum wells and bulk layers. But organic semiconductor microcavities have been studied only in the weak-coupling regime. Here we report an organic semiconductor microcavity that operates in the strong-coupling regime. We see characteristic mixing of the exciton and photon modes (anti-crossing), and a room-temperature vacuum Rabi splitting (an indicator of interaction strength) that is an order of magnitude larger than the previously reported highest values for inorganic semiconductors. Our results may lead to new structures and device concepts incorporating hybrid states of organic and inorganic excitons, and suggest that polariton lasing may be possible.