The MAMMOTH-I Nebula at redshift 2.3 is one of the largest known Ly \alpha nebulae in the Universe , spanning \sim 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 \sim 30 kpc into the circum-galactic medium ( CGM ) .
In the centermost region , the CO has a very low velocity dispersion ( FWHM _ { CO } \sim 85 km s ^ { -1 } ) , 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 _ { H 2 } \sim 1.4 ( \alpha _ { 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 .