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We discovered rapid intra-day variability in radio source PMN J1726+0639 at GHz frequencies, during a survey to search for such variability with the Australia Telescope Compact Array. Follow-up observations were conducted over two years and revealed a clear, repeating annual cycle in the rate, or characteristic timescale, of variability, showing that the observed variations can be attributed to scintillations from interstellar plasma inhomogeneities. The strong annual cycle includes an apparent "standstill" in April and another in September. We fit kinematic models to the data, allowing for finite anisotropy in the scintillation pattern. The cycle implies a very high degree of anisotropy, with an axial ratio of at least 13:1, and the fit is consistent with a purely one-dimensional scintillation pattern. The position angle of the anisotropy, and the transverse velocity component are tightly constrained. The parameters are inconsistent with expectations from a previously proposed model of scattering associated with plasma filaments radially oriented around hot stars. We note that evidence for a foreground interstellar cloud causing anomalous Ca II absorption towards the nearby star Rasalhague ($α$ Oph) has been previously reported, and we speculate that the interstellar scintillation of PMN 1726+0639 might be associated with this nearby cloud.