We have investigated exciton-photon coupling phenomena in a bulk CuCl microcavity with distributed Bragg reflectors consisting of ${\text{PbF}}_{2}$ and ${\text{PbBr}}_{2}$ layers prepared by vacuum deposition. Reflectance spectra observed at various incident-light angles demonstrate the strong-coupling behavior of the exciton and photon modes; namely, the energies of reflectance dips exhibit incident-angle dependence accompanied by an anticrossing behavior peculiar to the cavity polaritons. In addition, the emission from the lower polariton branch was detected with angle-resolved photoluminescence spectroscopy. From the phenomenological analysis with a $3\ifmmode\times\else\texttimes\fi{}3$ Hamiltonian of the cavity-polariton modes originating from the ${Z}_{3}$ exciton, ${Z}_{1,2}$ exciton, and photon, it has been revealed that the vacuum-Rabi-splitting energies are 97 and 162 meV for the ${Z}_{3}$ and ${Z}_{1,2}$ excitons, respectively. These giant Rabi-splitting energies result from the large oscillator strengths of the relevant excitons. Furthermore, the spectral profile of reflectance is reasonably explained by calculations based on a theory of nonlocal linear optical response.