  We theoretically study circular dichroism of chiral molecules embedded inside
a helicity-preserving Fabry-P\'erot cavity. We find an increase of the
intrinsic chiroptical response of the molecules by 2 orders of magnitude and
report the first clear signature of chiral cavity polaritons upon entering the
regime of strong light-matter coupling. We study a cavity design based on two
dielectric photonic crystal mirrors acting, in a narrow frequency range, as
efficient polarization cross-converters in transmission for one polarization
and almost perfect reflectors for the other polarization. We show that a
Pasteur medium hosted inside such a cavity can couple efficiently to both the
outside of the cavity and to the helicity-preserving mode, inheriting an
enhanced chiral character. We expect such a device to be useful in the future
to design ultrasensitive chiral sensors for optics and stereochemistry.
