  The present constraint on the space parity ($\mathcal{P}$) and time
reflection invariance ($\mathcal{T}$) violating electron electric dipole moment
($e$EDM) is based on the observation of the electron spin precession in an
external electric field using the ThO molecule. We propose an alternative
approach: observation of the $\mathcal{P}$,~$\mathcal{T}$-odd Faraday effect in
an external electric field using the cavity-enhanced polarimetric scheme in
combination with a molecular beam crossing the cavity. Our theoretical
simulation of the proposed experiment with the PbF and ThO molecular beams
shows that the present constraint on the $e$EDM in principle can be improved by
a few orders of magnitude.
