It has also been refused permission to appeal to the Supreme Court.
Weller and his wife won the damages last year after pictures of his three youngest children were "plastered" on Mail Online in 2012.
The couple complained that the un-pixellated shots were "plainly voyeuristic".
They sued Associated Newspapers, which publishes the Daily Mail, the Mail on Sunday and Metro, for misuse of private information on behalf of their daughter Dylan, who was 16 when the pictures appeared online, and twin sons John-Paul and Bowie, who were 10 months old.
Associated Newspapers appealed against the ruling because it said it created what was, in effect, an "image right" - that is, giving legal control to the subject of the photograph - for the first time and this would have far-reaching adverse effects on the freedom of the UK media.
However three High Court judges today dismissed the appeal and refused Associated Newspapers permission to appeal to the Supreme Court.
Seven paparazzi photos were published in October 2012 under the headline "A family day out: Paul Weller takes wife Hannah and his twin sons out for a spot of shopping in the hot LA sun".
A photographer had followed the family on a shopping trip in Santa Monica, California, and took photographs without their consent despite being asked to stop.
Lawyers for Mail Online had argued the publication of the images was entirely in line with the law in California where they were taken by a freelance photographer.
The judge agreed the images could have been published legally in California, but said their appearance in the UK violated the right to privacy enshrined in the European Convention on Human Rights.
Since their win at the High Court, Hannah Weller has campaigned to stop papers publishing photos of children without their parents' consent.
Associated Newspapers has lost its challenge to a High Court decision to award £10,000 privacy damages to rock star Paul Weller.