  We report on the first Green Bank Telescope observations at 427, 820 and 1400
MHz of the newly discovered, highly inclined and relativistic double pulsar
binary. We focus on the brief eclipse of PSR J0737-3039A, the faster pulsar,
when it passes behind PSR J0737-3039B. We measure a frequency-averaged eclipse
duration of 26.6 +/- 0.6 s, or 0.00301 +/- 0.00008 in orbital phase. The
eclipse duration is found to be significantly dependent on radio frequency,
with eclipses longer at lower frequencies. Specifically, eclipse duration is
well fit by a linear function having slope (-4.52 +/- 0.03) x 10^{-7}
orbits/MHz. We also detect significant asymmetry in the eclipse. Eclipse
ingress takes 3.51 +/- 0.99 times longer than egress, independent of radio
frequency. Additionally, the eclipse lasts (40 +/- 7) x 10^{-5} in orbital
phase longer after conjunction, also independent of frequency. We detect
significant emission from the pulsar on short time scales during eclipse in
some orbits. We discuss these results in the context of a model in which the
eclipsing material is a shock-heated plasma layer within the slower PSR
J0737-3039B's light cylinder, where the relativistic pressure of the faster
pulsar's wind confines the magnetosphere of the slower pulsar.
