  During the motion of a binary pulsar around the Galactic center, the pulsar
and its companion experience a wind of dark-matter particles that can affect
the orbital motion through dynamical friction. We show that this effect
produces a characteristic seasonal modulation of the orbit and causes a secular
change of the orbital period whose magnitude can be well within the astonishing
precision of various binary-pulsar observations. Our analysis is valid for
binary systems with orbital period longer than a day. By comparing this effect
with pulsar-timing measurements, it is possible to derive model-independent
upper bounds on the dark-matter density at different distances $D$ from the
Galactic center. For example, the precision timing of J1713+0747 imposes
$\rho_{\rm DM}\lesssim 10^5\,{\rm GeV/cm}^3$ at $D\approx7\,{\rm kpc}$. The
detection of a binary pulsar at $D\lesssim 10\,{\rm pc}$ could provide
stringent constraints on dark-matter halo profiles and on growth models of the
central black hole. The Square Kilometer Array can improve current bounds by 2
orders of magnitude, potentially constraining the local density of dark matter
to unprecedented levels.
