Circinus X-1 exhibited a bright X-ray flare in late 2013. Follow-up observations with Chandra and XMM-Newton from 40 to 80 days after the flare reveal a bright X-ray light echo in the form of four well-defined rings with radii from 5 to 13 arcminutes, growing in radius with time. The large fluence of the flare and the large column density of interstellar dust towards Circinus X-1 make this the largest and brightest set of rings from an X-ray light echo observed to date. By deconvolving the radial intensity profile of the echo with the MAXI X-ray lightcurve of the flare we reconstruct the dust distribution towards Circinus X-1 into four distinct dust concentrations. By comparing the peak in scattering intensity with the peak intensity in CO maps of molecular clouds from the Mopra Southern Galactic Plane CO Survey we identify the two innermost rings with clouds at radial velocity ~ -74 km/s and ~ -81 km/s, respectively. We identify a prominent band of foreground photoelectric absorption with a lane of CO gas at ~ -32 km/s. From the association of the rings with individual CO clouds we determine the kinematic distance to Circinus X-1 to be $D_{Cir X-1} = 9.4^{+0.8}_{-1.0}$ kpc. This distance rules out earlier claims of a distance around 4 kpc, implies that Circinus X-1 is a frequent super-Eddington source, and places a lower limit of $Gamma gtrsim 22$ on the Lorentz factor and an upper limit of $theta_{jet} lesssim 3^{circ}$ on the jet viewing angle.
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