We present X-ray imaging and spectroscopy of the redshift z=7.084 radio-quiet quasar ULAS J112001.48+064124.3 obtained with Chandra and XMM-Newton. The quasar is detected as a point source with both observatories. The Chandra observation provides a precise position, confirming the association of the X-ray source and the quasar, while a sufficient number of photons is detected in the XMM-Newton observation to yield a meaningful X-ray spectrum. In the XMM-Newton observation the quasar has a 2-10 keV luminosity of 4.7+-0.9 times 10^44 erg/s and a spectral slope alpha = 1.6+0.4/-0.3 (where f_nu is proportional to nu^-alpha). The quasar appears to have dimmed in the 15 months between the two observations, with a 2-10 keV luminosity of $1.8+1.0/-0.7 times 10^45 erg/s during the Chandra observation. We derive optical to X-ray spectral slopes alpha_OX of 1.76+-0.07 and 1.54+0.09/-0.08 at the times of the XMM-Newton and Chandra observations respectively, consistent with the range of alpha_OX found in other quasars of comparable ultraviolet luminosity. The very soft X-ray spectrum suggests that the quasar is accreting above the Eddington rate, L/L_Edd = 5+15/-4, compared to L/L_Edd = 1.2+0.6/-0.5 derived from the rest-frame ultraviolet. Super-Eddington accretion would help to reduce the discrepancy between the age of the quasar implied by the small size of the ionized near zone in which it sits (<10^7 years), and the characteristic e-folding time (2.5 times 10^7 years if L/L_Edd=2). Such super-Eddington accretion would also alleviate the challenging constraints on the seed black hole mass provided that the quasar has been rapidly accreting throughout its history. The remnant of an individual population III star is a plausible progenitor if an average L/L_Edd>1.46 has been maintained over the quasars lifetime.
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