The New Day, created by publishing giant Trinity Mirror, hit the newsstands on Monday, carrying a front-page story of a report into children who cared for their parents and a column by Cameron calling on Britain to remain in the European Union.
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Despite plummeting sales of print newspapers, which led The Independent to announce last week that it was moving online, the new paper insisted that its unique format would help it survive.
"We would of course be completely daft if what we were launching was just another newspaper. But it's not," said a message from its editorial team.
"Yes, we have news and we are made of paper. But that's pretty much where the similarities end. We know this can't just be another newspaper."
Barry Rabbetts, the paper's executive editor, tweeted an image of the front page, adding: "When you pick up @thenewdayuk tomorrow you'll see it's very different."
The first ever front page of The New Day newspaper @thenewdayuk #seizethenewday pic.twitter.com/EwLHnNw5Sj
— Barry Rabbetts (@BRabbetts) February 28, 2016
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Cameron wrote a one-page column warning of the unknowns of a Brexit, leading it by saying "It's a new day. A new day when three million people will go to work to do a job that is linked to Britain's trade with Europe."
As well as newspaper staples such as puzzles, showbiz gossip, recipes and horoscopes, the paper also carried reports into the plight of albino children in Africa and the "army" of British youngsters who have to look after their parents.
The paper, aimed at 35-55 year-olds, promises to be free from political bias, focus on good news as well as bad and to mix up traditionally male and female stories rather than having separate sections.
The 40-page newspaper was given away free on Monday, and will trial at 25 pence (0.32 euro, $0.35) for two weeks and 50 pence after that.
A cut-price version of The Independent, the "i", was launched in 2010 but Trinity Mirror said the new daily will be "an entirely new newspaper, not a sister title or light version" of the Daily Mirror, its main publication.
The Daily Mirror lost 12 per cent of sales year on year, down to just over 800,000 in January.
The new paper is being launched shortly after the Independent daily said it would be printing its last print edition on March 26 and shifting fully online.
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