  The violation of the weak equivalence principle (EP) in the gravitational
field of the Earth, described by the E\"otv\"os parameter $\eta_\oplus$, was
recently constrained to the level $\left|\eta_\oplus\right| \lesssim 10^{-14}$
by the MICROSCOPE space mission. The E\"otv\"os parameter $\eta_{\rm DM}$,
pertaining to the differential couplings of dark matter (DM) and ordinary
matter, was only tested to the level $\left| \eta_{\rm DM} \right| \lesssim
10^{-5}$ by the E\"ot-Wash group and lunar laser ranging. This test is limited
by the EP-violating driving force in the Solar neighborhood that is determined
by the Galactic distribution of DM. Here we propose a novel celestial
experiment using the orbital dynamics from radio timing of binary pulsars, and
obtain a competing limit on $\eta_{\rm DM}$ from a neutron star (NS) -- white
dwarf (WD) system, PSR J1713+0747. The result benefits from the large material
difference between the NS and the WD, and the large gravitational binding
energy of the NS. If we can discover a binary pulsar within $\sim 10$ parsecs
of the Galactic center, where the driving force is much larger in the expected
DM spike, precision timing will improve the test of the universality of free
fall towards DM and constrain various proposed couplings of DM to the Standard
Model by several orders of magnitude. Such a test probes the hypothesis that
gravity is the only long-range interaction between DM and ordinary matter.
