We report a series of observations of the interstellar scintillation (ISS) of the double pulsar J0737-3039 over the course of 18 months. As in earlier work, the basic phenomenon is the variation in the ISS caused by the changing transverse velocities of each pulsar, the ionized interstellar medium (IISM), and the Earth. The transverse velocity of the binary system can be determined both by very long baseline interferometry and timing observations. The orbital velocity and inclination is almost completely determined from timing observations, but the direction of the orbital angular momentum is not known. Since the Earth's velocity is known, and can be compared with the orbital velocity by its effect on the timescale of the ISS, we can determine the orientation Ω of the pulsar orbit with respect to equatorial coordinates (Ω = 65 ± 2°). We also resolve the ambiguity (i = 88.°7 or 91.°3) in the inclination of the orbit deduced from the measured Shapiro delay by our estimate i = 88.°1 ± 0.°5. This relies on the analysis of the ISS over both frequency and time, and provides a model for the location, anisotropy, turbulence level, and transverse phase gradient of the IISM. We find that the IISM can be well-modeled during each observation, typically of a few orbital periods, but its turbulence level and mean velocity vary significantly over the 18 months. © 2014. The American Astronomical Society. All rights reserved.