RMC already has permission to take fluorspar in the area. However, it had also been taking limestone - in breach of its planning permission - and has been served with an enforcement notice.
Local residents and countryside campaigners welcomed the decision. Peter Thompson, chairman of the Save Longstone Edge group, said: "We applaud the decision by the park authority in stopping the expansion of this massively destructive quarry. The decision means that even if the company appeals, they are prevented from further damaging the landscape during the lengthy appeals procedure ... They should accept the decision gracefully and pull out from Longstone Edge."
Elaine Gilligan, quarrying campaigner at Friends of the Earth, welcomed the decision but said: "The Government must change the law to allow the old minerals permissions in environmentally important areas to be revoked without local authorities having to pay huge sums in compensation."
The issue is seen as a test case by 11 national parks in England and Wales which face more than 100 similar proposals.
RMC which owns the site, said that it would appeal to the Deputy Prime Minister, John Prescott.
Lawyers for the National Park believe an inquiry could take up to eight months, with no guarantee that it would win. If Mr Prescott ruled in favour of the company, the planning board would have to compensate it for lost mining during that time.
Martin Doughty, a councillor and chairman of the National Park Authority, said: "This has been a very difficult decision. There is no question that we are required to protect the environment but, because of the complexities of the law and the interpretation of old planning permissions, this is easier said than done. We have had to prepare our case very carefully because we know any enforcement orders may be challenged by RMC. By issuing a stop notice, we could be liable for compensation."
The authority has already written to RMC warning them of its intention to issue the enforcement and stop notices. It will now explain in detail why it has rejected the company's proposals and put forward an alternative scheme for restoring the site.
Mr Doughty said: "RMC say they are concerned about the environment but, given the local and national outrage their actions have provoked, we hope they will accept that the original planning permission did not allow for the kind of extraction they are proposing."
David Bradley, of the Council for the Protection of Rural England, said: "We applaud this decisive action to protect the National Park. Last year we asked the Peak Park Authority to issue an enforcement notice to control the work. It seemed clear then that the existing planning consent only covered vein minerals - not limestone extraction.
He added that without the park authority's move, Backdale would have become "the biggest eyesore in the Peak District".