The broadening of atomic emission lines by high-velocity motion of gas near accreting supermassive black holes is an observational hallmark of quasars. Observations of broad emission lines could potentially constrain the mechanism for transporting gas inwards through accretion disks or outwards through winds. The size of this broad-line region has been estimated by measuring the light travel time delay between the variable nuclear continuum and the emission lines - a method known as reverberation mapping. In some models the emission lines arise from a continuous outflow, whereas in others they are produced by orbiting gas clouds. Directly imaging such regions has not hitherto been possible because of their small angular sizes (< 0.1 milli-arcseconds). Here we report a spatial offset (with a spatial resolution of ten micro-arcseconds or about 0.03 parsecs for a distance of 550 million parsecs) between the red and blue photo-centres of the broad Paschen-{alpha} line of the quasar 3C 273 perpendicular to the direction of its radio jet. This spatial offset corresponds to a gradient in the velocity of the gas and thus implies that the gas is orbiting the central supermassive black hole. The data are well fitted by a broad-line-region model of a thick disk of gravitationally bound material orbiting a black hole of 300 million solar masses. We infer a disk radius of 150 light days; a radius of 100-400 light days was found previously using reverberation mapping. The rotation axis of the disk aligns in inclination and position angle with the radio jet. Our results support the methods that are often used to estimate the masses of accreting supermassive black holes and to study their evolution over cosmic time.
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