We report the direct observation of 30-fs period Rabi oscillations between excitons and cavity photons in a strongly coupled $J$-aggregate microcavity by means of time-resolved up-conversion and spectral interferometry measurements. The time structure of the transmitted electric field, measured by linear spectral interferometry, shows pronounced ultrafast beats. Its spectral phase reveals a distinct signature caused by destructive interference between the coherent drive and the field radiated by the exciton. This antiresonance selectively probes the uncoupled exciton excitation, and its observation uncovers the coherent and ultrafast exchange of energy between the optically excited cavity and the $J$-aggregate excitons, as confirmed by transfer matrix calculations.