PSR J1906+0746 is a young radio pulsar ( \tau = 112 kyr , P = 144 ms ) in a tight binary ( P _ { orb } = 3.98 hr ) with a compact high-mass companion ( M _ { comp } \simeq 1.36 M _ { \odot } ) , at the distance of about 5 kpc . We observed this unique relativistic binary with the Chandra ACIS detector for 31.6 ks . Surprisingly , not a single photon was detected within the 3 ^ { \prime \prime } radius from the J1906+0746 radio position . For a plausible range of hydrogen column densities , n _ { H } = ( 0.5 – 1 ) \times 10 ^ { 22 } cm ^ { -2 } , the nondetection corresponds to the 90 % upper limit of ( 3– 5 ) \times 10 ^ { 30 } erg s ^ { -1 } on the unabsorbed 0.5–8 keV luminosity for the power-law model with \Gamma = 1.0 –2.0 , and \sim 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 ^ { \prime } centered on the pulsar . The measured 0.5–8 keV flux of the feature , \approx 3.1 \times 10 ^ { -14 } erg cm ^ { -2 } s ^ { -1 } , implies an unabsorbed luminosity of 1.2 \times 10 ^ { 32 } erg s ^ { -1 } ( 4.5 \times 10 ^ { -4 } of the pulsar ’ s \dot { E } ) for n _ { H } = 0.7 \times 10 ^ { 22 } cm ^ { -2 } . 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 .