The guitarist - who co-founded the band in 1976 - passed away following a short respiratory illness, with the news being announced via a statement posted on the group's official Twitter account.
The statement read: "This is so hard for us to write, but our great friend and Supreme Leader has died today.
"Andy's final tour in November was the only way he was ever really going to bow out; with a Stratocaster around his neck, screaming with feedback and deafening the front row. (sic)"
The statement - which is signed by bandmates John Sterry, Thomas McNeice and Tobias Humble - describes Gill as "one of the best to ever do it".
They said: "His uncompromising artistic vision and commitment to the cause meant that he was still listening to mixes for the upcoming record, whilst planning the next tour from his hospital bed.
"But to us, he was our friend - and we'll remember him for his kindness and generosity, his fearsome intelligence, bad jokes, mad stories and endless cups of Darjeeling tea. He just so happened to be a bit of a genius too.
"One of the best to ever do it, his influence on guitar music and the creative process was inspiring for us, as well as everyone who worked alongside him and listened to his music. And his albums and production work speak for themselves. Go give 'em a spin for him...
"Love you mate (sic)"