  Ferroelectricity, the electrostatic counterpart to ferromagnetism, has long
been thought to be incompatible with metallicity due to screening of electric
dipoles and external electric fields by itinerant charges. Recent measurements,
however, demonstrated signatures of ferroelectric switching in the electrical
conductance of bilayers and trilayers of WTe$_2$, a semimetallic transition
metal dichalcogenide with broken inversion symmetry. An especially promising
aspect of this system is that the density of electrons and holes can be
continuously tuned by an external gate voltage. This degree of freedom enables
investigation of the interplay between ferroelectricity and free carriers, a
previously unexplored regime. Here, we employ capacitive sensing in dual-gated
mesoscopic devices of bilayer WTe$_2$ to directly measure the spontaneous
polarization in the metallic state and quantify the effect of free carriers on
the polarization in the conduction and valence bands, separately. We compare
our results to a low-energy model for the electronic bands and identify the
layer-polarized states that contribute to transport and polarization
simultaneously. Bilayer WTe$_2$ is thus shown to be a canonical example of a
ferroelectric metal and an ideal platform for exploring polar ordering,
ferroelectric transitions, and applications in the presence of free carriers.
