We report the observation of the strong-coupling regime between organic excitons and confined photons within a microcavity structure. As the optically active organic semiconductors within our devices, we have used the organic dye tetra-(2,6-tert-butyl)phenol-porphyrin zinc [4TBPPZn] which has an intense narrow 'Soret' band absorption and also a cyanine dye which can form so-called 'J-aggregates'. Strong exciton-photon coupling is demonstrated from the anticrossing of exciton and photon modes. Room temperature vacuum-Rabi splittings more than one order of magnitude larger than the highest values reported for inorganic semiconductors are observed. Such results open the way to structures in which new hybrid states of organic and inorganic excitons can be expected.