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This paper presents optical R-band light curves and the time delay of the doubly imaged gravitationally lensed quasar SDSS J1001+5027 at a redshift of 1.838. We have observed this target for more than six years, between March 2005 and July 2011, using the 1.2-m Mercator Telescope, the 1.5-m telescope of the Maidanak Observatory, and the 2-m Himalayan Chandra Telescope. Our resulting light curves are composed of 443 independent epochs, and show strong intrinsic quasar variability, with an amplitude of the order of 0.2 magnitudes. From this data, we measure the time delay using five different methods, all relying on distinct approaches. One of these techniques is a new development presented in this paper. All our time-delay measurements are perfectly compatible. By combining them, we conclude that image A is leading B by 119:3 ± 3:3 days (1σ, 2.8% uncertainty), including systematic errors. It has been shown recently that such accurate time-delay measurements offer a highly complementary probe of dark energy and spatial curvature, as they independently constrain the Hubble constant. The next mandatory step towards using SDSS J1001+5027 in this context will be the measurement of the velocity dispersion of the lensing galaxy, in combination with deep Hubble Space Telescope imaging. [ABSTRACT FROM AUTHOR]