The high-redshift radio galaxy MRC 1138-262 ( ‘ Spiderweb Galaxy ’ ; z = 2.16 ) , is one of the most massive systems in the early Universe and surrounded by a dense ‘ web ’ of proto-cluster galaxies . Using the Australia Telescope Compact Array , we detected CO ( 1-0 ) emission from cold molecular gas – the raw ingredient for star formation – across the Spiderweb Galaxy . We infer a molecular gas mass of M _ { H 2 } = 6 \times 10 ^ { 10 } M _ { \odot } ( for M _ { H 2 } /L ’ _ { CO } = 0.8 ) . While the bulk of the molecular gas coincides with the central radio galaxy , there are indications that a substantial fraction of this gas is associated with satellite galaxies or spread across the inter-galactic medium on scales of tens of kpc . In addition , we tentatively detect CO ( 1-0 ) in the star-forming proto-cluster galaxy HAE 229 , 250 kpc to the west . Our observations are consistent with the fact that the Spiderweb Galaxy is building up its stellar mass through a massive burst of widespread star formation . At maximum star formation efficiency , the molecular gas will be able to sustain the current star formation rate ( SFR \approx 1400 M _ { \odot } yr ^ { -1 } , as traced by Seymour et al . ) for about 40 Myr . This is similar to the estimated typical lifetime of a major starburst event in infra-red luminous merger systems .