Most Rev Justin Welby made the comments in a private letter to a mother whose three sons were abused at their CofE school.
He was responding to the concerns of Marilyn Hawes, who had written several times to senior church officials and only received a generic response.
Her three sons suffered abuse at the hands of a headteacher who was later sentenced to 15 months in prison.
The churchgoer said that, despite her children going on to achieve 'great things', they hadn't received any support from the Church.
In his response, Archbishop Justin said: "I read your story with the same deep sense of sadness and dismay that I have felt on far too many other, similar accounts.
"The betrayal of Christ in such behaviour is complete; the church's failure to face the misdeeds of those in its service is inexcusable.
"I can only apologise for what happened then, and for what has happened now, most sincerely and with deep sorrow.
"It is now clear that in a huge number of institutions and localities, the abuse of children and vulnerable adults has been rampant."
His letter comes just weeks after the Church announced that clergy would be given more training on safeguarding to combat child sex abuse.
The mother told the Daily Mail she was "delighted" that an archbishop was finally dealing with the issue of abuse.