Carver, a long-serving coach at St James's Park, was put in charge for the second half of the season after Alan Pardew left for Crystal Palace.
The team won only three games under his stewardship, however, and were in danger of relegation from the Premier League until the final day of the campaign.
Stone, a former England international, had been a coach at the club since 2010.
(Reporting by Steve Tongue, editing by Ed Osmond)
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