  The double pulsar J0737-3039 is the only known system in which the
relativistic wind emitted by a radio pulsar demonstrably interacts with the
magnetosphere of another one. We report radio interferometric observations of
the J0737-3039 system with the VLA at three wavelengths, with each observation
spanning a full binary orbit. We detect J0737-3039 at 1.6 and 4.8 GHz, derive a
spectral index of -2.3 +/- 0.2, and place an upper limit on its flux density at
8.4 GHz. Orbital modulation is detected in the 1.6 GHz data with a significance
of ~2 sigma. Both orbital phase-resolved and phase-averaged measurements at 1.6
GHz are consistent with the entire flux density arising from the pulsed
emission of the two pulsars. Contrary to prior results, we find no evidence for
unpulsed emission, and limit it to less than 0.5 mJy (5 sigma).
