  PSR J1906+0746 is a 112-kyr-old radio pulsar in a tight relativistic binary
with a compact high-mass companion, at the distance of about 5 kpc. We observed
this unique system with the Chandra ACIS detector for 31.6 ks. Surprisingly,
not a single photon was detected within the 3" radius from the J1906+0746 radio
position. For a plausible range of hydrogen column densities,
n_H=(0.5-1)\times10^{22} cm^{-2}, the nondetection corresponds to the 90% upper
limit of (3-5)\times10^{30} erg s^{-1} on the unabsorbed 0.5-8 keV luminosity
for the power-law model with Gamma=1.0-2.0, and ~10^{32} erg s^{-1} on the
bolometric luminosity of the thermal emission from the NS surface. The inferred
limits are the lowest known for pulsars with spin-down properties similar to
those of PSR J1906+0746. We have also tentatively detected a puzzling extended
structure which looks like a tilted ring with a radius of 1.6' centered on the
pulsar. The measured 0.5-8 keV flux of the feature, 3.1\times10^{-14} erg
cm^{-2} s^{-1}, implies an unabsorbed luminosity of 1.2\times10^{32} erg s^{-1}
(4.5\times10^{-4} of the pulsar's spin-down power). Although all conventional
interpretations of the ring appear to be problematic, the pulsar-wind nebula
with an unusually underluminous pulsar remains the most viable interpretation.
