The double pulsar system J0737-3039 is not only a test bed for general relativity and theories of gravity, but also provides a unique laboratory for probing the relativistic winds of neutron stars. Recent X-ray observations have revealed a point source at the position of the J0737-3039 system, but have failed to detect pulsations or orbital modulation. Here we report on Chandra X-Ray Observatory High Resolution Camera observations of the double pulsar. We detect deeply modulated, double-peaked X-ray pulses at the period of PSR J0737-3039A, similar in appearance to the observed radio pulses. The pulsed fraction is ～ 70%. Purely nonthermal emission from pulsar A plausibly accounts for our observations. However, the X-ray pulse morphology of A, in combination with previously reported spectral properties of the X-ray emission, allows the existence of both nonthermal magnetospheric emission and a broad sinusoidal thermal emission component from the neutron star surface. No pulsations are detected from pulsar B, and there is no evidence for orbital modulation or extended nebular structure. The absence of orbital modulation is consistent with theoretical expectations of a Poynting-dominated relativistic wind at the termination shock between the magnetosphere of B and the wind from A and with the small fraction of the energy outflow from A intercepted by the termination shock.
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