He also said that the best estimate they now have is that between 135 and 270 women's lives were "shortened as a result" of the errors.
Women in England between the ages of 50 and 70 are now automatically invited for breast cancer screening every three years.
"Since 2009, in Northern Ireland women become eligible to be invited for breast screening every three years from the calendar year they turn 50 to the calendar year they turn 70".
A helpline will be set up to help women aged over 75 and talk them through the pros and cons of having breast screening - scans in older women sometimes pick up cancers which do not require treatment.
He said about 450,000 women aged between 68 and 71 weren't given a chance to have a mammogram.
The starkest numbers in Jeremy Hunt's statement were the deaths - between 135 and 270 women may have died early as a result of missed screening.
Of those women, Hunt said experts estimated up to 270 might have died prematurely.
Samia al Qadhi, chief executive of charity Breast Cancer Now, added: "Hundreds of thousands of women across England have been failed by this appalling error and some have had their lives shortened as a result". There are unlikely to be more fatalities than that, and the number could be less, he says.
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He said a computer algorithm failure was to blame, which meant women who had just turned 70 were not sent an invitation for a final scan as they should have been.
Mr Hunt said: "For them and others it is incredibly upsetting to know that you did not receive an invitation for screening at the correct time and totally devastating to hear you may have lost or be about to lose a loved one because of administrative incompetence".
All women who were not sent an invitation for their final screening will be given the opportunity to have a new appointment.
"Ultimately, we need funding for more training posts for radiologists to ensure the screening programme - and the NHS as a whole - has the vital imaging doctors it needs", said Caroline Rubin, vice president for clinical radiology at The Royal College of Radiologists.
Hunt apologised "wholeheartedly and unreservedly" for the suffering caused.
The 309,000 women thought to be affected by the issue will be contacted by the NHS this week.
Ms Thomas said: 'It's absolutely critical we understand what happened and make sure this situation never happens to another person again.