The Guardian had claimed this week that the Sun hacked into Brown's son Fraser's medical records before reporting he had cystic fibrosis. The Sun said its source was the father of another child with cystic fibrosis who heard the news "on the grapevine".
"In fact the information came from a different source and the Guardian apologises for its error," the paper wrote in its corrections column.
The Sun newspaper had strongly denied accessing Brown's family medical records without his knowledge. Brown was chancellor of the exchequer in 2006.
The paper said its 2006 story came from a member of the public, and released a video interview with the unidentified man. Brown said he was left "in tears" when he was told that the Sun was publishing the story.
He said he did not know how the newspaper got access to the details but said: "I can't think of any way that the medical condition of a child can be put into the public arena legitimately unless the doctor makes a statement or the family makes a statement."
The paper said the man's own child also had cystic fibrosis and he wanted to raise awareness of the condition.
The Sun, published by Rupert Murdoch''s News International, said it had conducted an inquiry into the allegations and that it had not accessed the medical records of Brown's son or commissioned anyone to do so.