We have investigated the eclipse of the 23-ms pulsar J0737-3039A by its 2.8-s companion PSR J0737-3039B in the recently discovered double pulsar system using data taken with the Green Bank Telescope at 820 MHz. We find that the pulsed flux density at eclipse is strongly modulated with half the periodicity of the 2.8-s pulsar. The eclipse occurs earlier and is deeper at those rotational phases of B when its magnetic axis is aligned with the line of sight than at phases when its magnetic axis is at right angles to the line of sight. This is consistent with the eclipse of A being due to synchrotron absorption by the shock-heated plasma surrounding B, the asymmetry arising from the higher plasma densities expected in the B magnetospheres polar cusps.
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