Intensity scintillations of radio pulsars are known to originate from interference between waves scattered by the electron density irregularities of interstellar plasma, often leading to parabolic arcs in the two-dimensional power spectrum of the recorded dynamic spectrum. The degree of arc curvature depends on the distance to the scattering plasma and its transverse velocity with respect to the line of sight. We report the observation of annual and orbital variations in the curvature of scintillation arcs over a period of 16 yr for the bright millisecond pulsar, PSR J0437-4715. These variations are the signature of the relative transverse motions of Earth, the pulsar, and the scattering medium, which we model to obtain precise measurements of parameters of the pulsar's binary orbit and the scattering medium itself. We observe two clear scintillation arcs in most of our >5000 observations, and we show that they originate from scattering by thin screens located at distances D 1 = 89.8 ± 0.4 pc and D 2 = 124 ± 3 pc from Earth. The best-fit scattering model we derive for the brightest arc yields the pulsar's orbital inclination angle, i = 137.°1 ± 0.°3, and longitude of ascending node, Ω = 206.°3 ± 0.°4. Using scintillation arcs for precise astrometry and orbital dynamics can be superior to modeling variations in the diffractive scintillation timescale, because the arc curvature is independent of variations in the level of turbulence of interstellar plasma. This technique can be used in combination with pulsar timing to determine the full three-dimensional orbital geometries of binary pulsars and provides parameters essential for testing theories of gravity and constraining neutron star masses.