Due to play the Carling Stage, The Organ said via their website: "It is with deep regret that we have to cancel our performances at the Reading and Leeds festivals due to severe exhaustion which has created a medical problem for one of our band members.
"We want to assure our audiences that we are not breaking up, but that time is needed for recovery."
Meanwhile, Paramore will be replaced by Sonic Boom Six after cancelling their appearances on the Lock-up Stage.
Singer Hayley Williams has been told to rest her voice after suffering a sore throat.
She said: "I was really hoping I could make it through the first leg of our US tour and get over for Reading and Leeds, but due to the problems I am having with my voice, we needed to take this week off so I can heal.
"We are really sorry for the change of plans, especially with such late notice. We will be back in October for our first overseas headlining tour and we look forward to making it up to you."