The decision, confirmed in details of the coalition agreement announced today, will anger airport operator BAA and London businesses.
But the news that expansion of the airport, which also included a sixth terminal, was being abandoned will delight residents groups, conservationists and local councils around the west London airport.
The coalition agreement also confirmed today that the new Government will rule out any extra runways at Stansted and Gatwick airports as well.
Heathrow operator BAA, supported by airlines and commerce in the capital, had long argued for a third runway.
Such a runway was envisaged in the government's 2003 aviation White Paper and the green light was given by the then Transport Secretary Geoff Hoon in January 2009.
Labour continued to support the idea in the face of fierce opposition and despite the fact that the village of Sipson would effectively be "lost" to the new runway.
The Tories opposed the plan, preferring improved rail links to and from Heathrow, including a direct link to a new London to Scotland high-speed rail line.
BAA and business leaders had said Heathrow needed extra capacity and that without it London - and consequently the UK - would lose out to rival European cities as a place to visit and do business.
Ben Stewart, of Greenpeace, said the scrapping of the third runway at Heathrow was "fantastic news that will be met with great relief across west London".
He went on: "For years the village of Sipson has faced the prospect of destruction but now, finally, it looks likes that threat has been lifted once and for all.
"A third runway at Heathrow was always a bizarre proposal that made no sense to anybody who understood the impact aviation has on our climate.
"The politicians who promised they would do this have been good to their word, and for that they should be thanked. Corks will be popping for a few days after this announcement, though probably not at BAA headquarters."
Wandsworth in south west London was one of six councils which had mounted a legal challenge to the Labour Government's Heathrow plans.
Wandsworth Council leader Edward Lister said today: "Today's news is the final proof of the extraordinary success of our campaign. First we won in the courts and now we've got the new Government confirming that it's all over.
"This won't be the end of our campaigning. We will want to ensure that the current relief offered to residents by runway (take-off and landing) alternation is maintained and we will be pressing for reductions in night flights and more stringent noise and air quality controls.
"The new Government now has the chance to draw a line under an unsatisfactory period of public administration when too often it seemed that the aviation lobby's interests were being placed ahead of the concerns of local people about the environment and their quality of life."
Actor Richard Briers said he would be going to 10 Downing Street tomorrow to present David Cameron and Nick Clegg with a legal Deed of Trust containing the names of more than 90,000 people who jointly own a plot of land at the centre of the proposed runway development.
Briers, who said he would be thanking the two men for scrapping the third runway plan, added: "There are so many people named on this deed. It's terribly heavy, but I'll give it to the new Prime Minister and his Deputy."
John Stewart, chairman of Heathrow residents' group Hacan, said, "The third runway is now dead in the water. It is good news for London. A third runway would have been bad for the capital's environment and was not essential for its economic well-being. A long campaign has ended in victory. We are delighted."
Geraldine Nicholson, chairman of the No Third Runway Action Group, said: "It is a great decision. It means that thousands of people will not be forced out of their homes to make way for a new runway.
"Our task now is to regenerate Sipson and the other villages which have lived with so much uncertainty in recent years."
A BAA spokesman said: "We will work with the new Government to ensure that airports policy provides the strong international trading connections on which the UK's jobs and future competitiveness depend."
Dr Adam Marshall, director of policy at the British Chambers of Commerce, said: "As one of the world's premier business cities, London requires one of the world's premier hub airports.
"By putting forward a policy that constrains Heathrow and aviation capacity throughout the South East of England, this decision will diminish the region's attractiveness to investors - with potential consequences for the UK economy as a whole."
Friends of the Earth's executive director, Andy Atkins, said: "We're delighted that the new Government has scrapped plans to expand UK airports. This is an encouraging sign that the coalition takes cutting aviation emissions seriously. They must now rule out expansion at regional airports too.
"We need a new aviation strategy which makes carbon reduction a priority and goes at least as far as Labour's target to limit aviation emissions to 2005 levels by 2050."
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