'It is also clear that there is a very significant legacy of unacknowledged cases in the Church of England. We are taking all necessary steps to face these.'
Marilyn Hawes says she had tried and failed on several occasions to get an apology from the church, before finally getting a response from Welby.
Ms Hawes, a former music teacher who now runs an anti-abuse charity, first wrote to Rowan Williams in 2003 after her sons' abuser was convicted. The headmaster was sentenced to 15 months in prison.
She wrote: 'I have been an active church-goer my whole life - 50 years - and for the first time I am doubting, not my faith, but the sincerity, value and integrity of a church community.'
She received a two-paragraph, routine reply from an official, but decided to try to contact the church again after Welby was appointed as archbishop of Canterbury last year.
She wrote that the church community 'abandoned' her, saying: 'I went on to have a nervous breakdown and suicidal thoughts. People would cross the road or exit the Post Office rather than face me. Nobody from the church cared.'
Her sons, now adults, had 'achieved great things', she wrote, and had 'overcome the sexual abuse'.
However, she added, 'Nobody in any church has shown them any active help.'
Welby's comments come as the CoE plans to roll out training on safeguarding children for all its clergy.
Tim Loughton, former children's minister, said: 'The new archbishop should be congratulated for clearly focussing on the wrongs of the past throughout the church and his determination that every effort should be made to prevent them again in the future.'