The violation of the weak equivalence principle (EP) in the gravitational field of Earth, described by the Eötvös parameter η_{⊕}, was recently constrained to the level |η_{⊕}|≲10^{-14} by the MICROSCOPE space mission. The Eötvös parameter η_{DM}, pertaining to the differential couplings of dark matter (DM) and ordinary matter, was only tested to the level |η_{DM}|≲10^{-5} by the Eöt-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 η_{DM} from a neutron-star-white-dwarf (NS-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 ∼10 pc 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.