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A unique test of a cornerstone of general relativity, the gravitational redshift effect, was conducted with the space radio telescope RadioAstron, which has been in a highly eccentric orbit around the Earth since 2011. We probed the flow of time on board, which varies while RadioAstron passes through the varying gravitational potential of the Earth, by recording the spacecraft’s downlink signal, which is synchronized to its ultra-stable on-board H-maser, at the ground radio telescopes. The experiment requires us to extract the tiny gravitational redshift signal from the mixture of a number of much larger effects, including those of the nonrelativistic Doppler frequency shift, signal propagation through the Earth’s troposphere and ionosphere, higher-order kinematic effects of special relativity, Earth tides, and various instrumental effects. A total of 17 successful experiments were performed while the on-board H-maser was operational. Preliminary data analysis of the data from two experiments gives us a fractional accuracy of the gravitational redshift test of at least 2x10^{-4}, which is comparable to that obtained by Gravity Probe A. We present arguments in favour of the experiment uncertainty currently being dominated by that of the ionospheric frequency shift and discuss prospects for improving the achieved accuracy further.