The broadening of atomic emission lines by high-velocity motion of gas near accreting supermassive black holes is an observational hallmark of quasars1. Observations of broad emission lines could potentially constrain the mechanism for transporting gas inwards through accretion disks or outwards through winds2. The size of regions for which broad emission lines are observed (broad-line regions) has been estimated by measuring the delay in light travel time between the variable brightness of the accretion disk continuum and the emission lines3-a method known as reverberation mapping. In some models the emission lines arise from a continuous outflow4, whereas in others they arise from orbiting gas clouds5. Directly imaging such regions has not hitherto been possible because of their small angular size (less than 10-4 arcseconds3,6). Here we report a spatial offset (with a spatial resolution of 10-5 arcseconds, or about 0.03 parsecs for a distance of 550 million parsecs) between the red and blue photo-centres of the broad Paschen-α 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 3 × 108 solar masses. We infer a disk radius of 150 light days; a radius of 100-400 light days was found previously using reverberation mapping7-9. 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.