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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.