Herein, we have measured the phase dispersion of a cavity polariton in a single-crystalline anthracene microcavity via interference spectroscopy. Results show slow-light dispersion at the cavity polariton's lower polariton resonance, which agrees well with the transfer matrix method calculation results. According to the theoretical calculations, the slow-light dispersion's group refractive index is 24. We found that the structural dispersion due to a phenomenological photon cavity and material dispersion caused by the exciton contributed to the large group refractive index. These efforts will aid in realizing optical buffer memories for organic microcavities.