The MAMMOTH-I Nebula at redshift 2.3 is one of the largest known Ly-alpha nebulae in the Universe, spanning ~440 kpc. Enormous Ly-alpha nebulae like MAMMOTH-I typically trace the densest and most active regions of galaxy formation. Using sensitive low-surface-brightness observations of CO(1-0) with the Very Large Array, we trace the cold molecular gas in the inner 150 kpc of the MAMMOTH-I Nebula. CO is found in four regions that are associated with either galaxies or groups of galaxies that lie inside the nebula. In three of the regions, the CO stretches up to ~30 kpc into the circum-galactic medium (CGM). In the centermost region, the CO has a very low velocity dispersion (FWHM$_{rm CO}$ ~ 85 km/s), indicating that this gas is dynamically cold. This dynamically cold gas coincides with diffuse restframe optical light in the CGM around a central group of galaxies, as discovered with the Hubble Space Telescope. We argue that this likely represents cooling of settled and enriched gas in the center of MAMMOTH-I. This implies that the dynamically cold gas in the CGM, rather than the obscured AGN, marks the core of the potential well of this Ly-alpha nebula. In total, the CO in the MAMMOTH-I Nebula traces a molecular gas mass of M$_{rm H2}$ ~ 1.4 ($alpha_{rm CO}$/3.6) $times$ 10$^{11}$ M$_{odot}$, with roughly 50% of the CO(1-0) emission found in the CGM. Our results add to the increasing evidence that extended reservoirs of molecular gas exist in the CGM of massive high-z galaxies and proto-clusters.
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