  We propose a novel experiment to search for dark matter, based on the
application of an electric field inside a microwave cavity and electrometry
using Rydberg atoms. We show that this kind of experiment could be extremely
useful for detecting specific dark matter candidates, namely massive vector
fields coupled to the photon field, more commonly known as dark photons. Such a
massive vector field is a good candidate for dark matter. Using realistic
experimental parameters we show that such an experiment could improve the
current constraint on the coupling constant of the dark photons to Standard
Model photons in the 1~$\mu$eV to 10~$\mu$eV mass range, with the possibility
of tuning the maximum sensitivity via the cavity size. The main limiting
factors on the sensitivity of the experiment are the amplitude stability of the
applied field and the measurement uncertainty of the electric field by the
atoms.
