  SPT0311-58 is the most massive infrared luminous system discovered so far
during the Epoch of Reionization (EoR). In this paper, we present a detailed
analysis of the molecular interstellar medium at z = 6.9, through
high-resolution observations of the CO(6-5), CO(7-6), CO(10-9), [CI](2-1), and
p-H2O(211-202) lines and dust continuum emission with the Atacama Large
Millimeter/submillimeter Array (ALMA). The system consists of a pair of
intensely star-forming gravitationally lensed galaxies (labelled West and
East). The intrinsic far-infrared luminosity is (16 $\pm$ 4)$\times\rm 10^{12}
\ \rm L_{\odot}$ in West and (27 $\pm$ 4)$\times\rm 10^{11} \ \rm L_{\odot}$ in
East. We model the dust, CO, and [CI] using non-local thermodynamic equilibrium
radiative transfer models and estimate the intrinsic gas mass to be (5.4 $\pm$
3.4)$\times\rm 10^{11} \ \rm M_{\odot}$ in West and (3.1 $\pm$ 2.7)$\times\rm
10^{10} \ \rm M_{\odot}$ in East. We find that the CO spectral line energy
distribution in West and East are typical of high-redshift sub-millimeter
galaxies (SMGs). The CO-to-H2 conversion factor ($\alpha_{CO}$) and the gas
depletion time scales estimated from the model are consistent with the
high-redshift SMGs in the literature within the uncertainties. We find no
evidence of evolution of depletion time with redshift in SMGs at z > 3. This is
the most detailed study of molecular gas content of a galaxy in the EoR
to-date, with the most distant detection of H2O in a galaxy without any
evidence for active galactic nuclei in the literature.
