Wales' health minister Vaughan Gething has said he is sorry that 13,000 letters for people most vulnerable to coronavirus were mistakenly sent to the wrong addresses.
More than 80,000 people are meant to get a "shielding letter".
The Information Commissioner's Office (ICO) says it is looking into the data breach, after the Welsh NHS referred itself to the regulator.
Plaid Cymru said it was a "potentially disastrous mistake".
Mr Gething said the letters had now been re-sent "so people should receive them within the next day or two".
The ICO, which can fine organisations for not protecting private information, said it would bear in mind the current crisis in its response.
BBC Wales has been told local government minister Julie James briefed council leaders about the error on Friday.
The letters from the chief medical officer, which should have been delivered by 3 April, are for those with serious underlying health conditions and advise people to stay at home for 12 weeks.
They contain information and advice, including how those who have no-one else to support them can get medication and other essential items like food.
Those eligible for the letter qualify for priority delivery slots from supermarkets, although there have been delays introducing that system here.
The Welsh Government said local authorities and supermarkets had been provided with the correct addresses from the outset, with the latter using these for delivery slots.
On Wednesday, Mr Gething told BBC Radio Wales Breakfast with Oliver Hides the data had been processed at "remarkable speed".
"In one copy of that data the previous address was used rather than the current one," he said.
"I am sorry the error happened, and I know that there were people who will be worried about it.
"But all those letters have now gone out so people should receive them within the next day or two."
Plaid Cymru's local government spokeswoman Delyth Jewell said: "This is a potentially disastrous mistake which could needlessly endanger lives."
She called for details of "what went wrong for this to happen in the first place".
Paul Davies, who leads the Conservative group in Cardiff Bay, said the error was "just unacceptable".
"How on earth did 13,000 letters out of a total of 80,000 go to the wrong addresses?" he tweeted.
"We should not see a mistake like this on such a massive scale."
Earlier, a Welsh Government spokeswoman said: "Due to a processing error within the NHS Wales Informatics Service (NWIS), some shielding letters were posted to a previous address, where the intended recipient had recently moved.
"All letters have since been reissued to the correct addresses.
"We provided all local authorities with the correct details from the start of this process, and they have been directly contacting each person involved over the past two weeks."
She added: "We fully understand the concern this would have caused people and sincerely apologise for the mistake."
An ICO spokesperson said: "NHS Wales Informatics Services has reported a data breach to us and we will be making enquiries.
"People have a right to expect their personal information be protected.
"We'll bear in mind the current crisis when assessing breach reports such as this, taking an appropriately proportionate approach."