A member of staff probably spoke "without authorisation" about the medical condition of Gordon Brown's son prior to publication of a 2006 story in The Sun, a health boss said on Monday.
But officials at NHS Fife did not think that there had been "inappropriate access" to medical records relating to Fraser Brown - who was born in 2006, he added.
John Wilson, chief executive of NHS Fife, issued a statement after Mr Brown criticised the story about his son's cystic fibrosis at the Leveson Inquiry.
Mr Brown had complained about the story in July 2011 and The Sun denied accessing medical records.
A letter seen by Lord Justice Leveson revealed that Mr Brown was told of the result of health bosses' investigations last week.
The letter - written by Mr Wilson to Mr Brown on June 7 - was posted on the Leveson Inquiry website on Monday evening.
In the letter, Mr Wilson says, "we believe that there was no inappropriate access to the child's medical records".
And Mr Wilson tells Mr Brown: "We now accept that it is highly likely that, sometime in 2006, a member of staff in NHS Fife spoke, without authorisation, about the medical condition of your son, Fraser."
In an editorial, The Sun said the story "originated from a member of the public whose family has also experienced cystic fibrosis. He came to The Sun with this information voluntarily because he wanted to highlight the cause of those afflicted by the disease".
But Mr Brown today questioned The Sun's explanation, telling Lord Justice Leveson: "In 2006, The Sun claimed they had a story from a man in the street who happened to be the father of someone who suffered from cystic fibrosis. I never believed that could be correct."
Mr Wilson said in his statement that it was "highly likely" that a member of staff "spoke without authorisation".
"We now accept that it is highly likely that, sometime in 2006, a member of staff in NHS Fife spoke, without authorisation, about the medical condition of Mr Brown's son, Fraser," said Mr Wilson.
"With the passage of time it has not been possible to identify all the circumstances.
"We believe, however, that there was no inappropriate access to the child's medical records."
He said patent confidentiality procedures had been tightened and added: "I have apologised to Mr and Mrs Brown and we have taken steps to ensure that what happened to Mr and Mrs Brown and their family should not happen again."
Sun publisher News International welcomed NHS Fife's findings on Fraser's medical records.
"We welcome the fact that NHS Fife have today said that they believe there was 'no inappropriate access' to the medical records of Gordon Brown's son," said a spokesman. "The Sun stands by previous statements issued on the matter."
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