  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 non-thermal 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 non-thermal 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.
