The 33-year-old suffered the problem earlier in the week and is now set to be replaced by Liverpool's Jordan Henderson ahead of Saturday's friendly against Belgium
The 33-year-old was sent for a scan on Thursday, with the Football Association confirming the news that he will miss the tournament.
Liverpool midfielder Jordan Henderson will now be called up, subject to approval by Uefa following Wednesday's deadline to name a 23-man squad for the tournament.
Lampard had been set to feature in his fourth major tournament this summer, but he will now follow Gareth Barry, who suffered a stomach injury last weekend, out of England's training camp.
"I’m especially disappointed for Frank himself. He has been in very good form and was looking forward to the tournament"
- Roy Hodgson
“It’s a huge blow for the team, especially coming after the loss of another senior player in Gareth Barry," he said.
"The fact that the doctor cannot guarantee Frank could take part in any of the group matches means we have to replace him in the central midfield area.
“Apart from it being a huge blow to me and the team, I’m especially disappointed for Frank himself. He has been in very good form and was looking forward to the tournament. He made a huge contribution towards us qualifying for the Euros so it is cruel to be forced out through injury just before the tournament.”
England's last warm-up game will be against Belgium on June 2 ahead of their opening Group D fixture against France on June 11.