The child's mother Helen Macintyre, who had an affair with married father-of-four Johnson, had attempted to prevent any publication of her daughter Stephanie's identity or link to Johnson.
Rumours about Stephanie's paternity surfaced shortly after she was born in November 2009. Neither Johnson nor Macintyre, who was also married to property developer Pierre Rolin, has ever discussed the affair, confirmed or denied Johnson is the father of Stephanie.
Johnson's wife Marina Wheeler threw him out of the family home after learning about his affair.
Macintyre was appealing against a judge's decision in July 2012 not to award damages for details of the affair being published and the refusal to grant an injunction preventing reprinting of the story.
High Court judge Mrs Justice Nicola Davies upheld the ruling that the Associated Press, publishers of the Daily Mail, were justified in publishing the story.
Associated Newspapers was then ordered to pay £15,000 in privacy damages for publishing photographs of Stephanie, but the paper will have 80% of its legal costs paid by the appellant, around £200,000, according to the Mail.
Her appeal against the ruling was rejected by Master of the Rolls Lord Justice Dyson, who said: "It is not in dispute that the legitimate public interest in the father’s character is an important factor to be weighed in the balance against the child’s expectation of privacy."
"It is fanciful to expect the public to forget the fact that a man who is said to be the baby’s father, and who is a major public figure, has fathered a child after a brief adulterous affair (not for the first time).
"The mother accepted in cross-examination that any woman who embarked on an affair with the father was “playing with fire” and that such an affair was bound to attract “very considerable media attention”."Suggest a correction