We present numerical calculations of the photon-light-pseudoscalar-boson (LPB) production in the recently discovered binary pulsar system J0737-3039. Light pseudoscalar bosons oscillate into photons in the presence of strong magnetic fields. In the context of this binary pulsar system, this phenomenon attenuates the light beam emitted by one of the pulsars, when the light ray goes through the magnetosphere of the companion pulsar. We show that such an effect is observable in the gamma-ray band since the binary pulsar is seen almost edge-on, depending on the values of the LPB mass and on the strength of its two-photon coupling. Our results are surprising in that they show a very sharp and significant (up to 50%) transition probability in the gamma-ray (> tens of MeV) domain. The observations can be performed by the upcoming NASA GLAST mission.