A councilor in Conway said he received the text with the results of a coronavirus test of 20 people who were “shocked and disappointed”.
Adrian Mason of Calvin Bay Town Council said he had never tested for the virus and did not understand how his mobile number was found.
The test results came in separate batches over a week and included names and dates of birth.
The Department of Health and Social Services said it would investigate.
Mr Mason, who represents Rue to the town council and also a lawyer, said: “People trust the NHS with their information.
“If I got this kind of information, who else?”
He informed his local MP, David Jones, of the violation and the two are now calling for an investigation.
“It’s in the public interest,” Mr Mason said.
“If they have sent me this message, are they not waiting for those who are waiting for them?”
“One of the results was a one-day-old baby and his father, it’s the worst.”
A spokesman for the Department of Health and Social Services said: “Everyone tested positive is contacted directly and informed of their results via text or email.”
They said the department would investigate the matter.
Mr Mason said he had not yet been contacted directly by the Department of Health and Social Care or the Information Commissioner’s Office, which was also aware of the error.
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