Francesco Schettino had handed himself in to the Rebibbia prison in Rome after the verdict, his lawyer said.
He was sentenced in 2015 after a court found him guilty of manslaughter, causing a maritime accident and abandoning ship.
The Costa Concordia cruise ship capsized after hitting rocks off the Tuscan island of Giglio.
Francesco Schettino was nicknamed “Captain Coward” by the media, after the coastguard released recordings of him in a lifeboat resisting orders to return to the stricken vessel.
More than 4,000 people were aboard at the time and were forced into a chaotic evacuation.
Prosecutors say Francesco Schettino steered too close to the island to show off to a dancer, Domnica Cemortan, who was with him at the helm.
However, he blamed communication problems with the Indonesian helmsman.
The court ruling was welcomed by a lawyer representing relatives of the victims, who said it represented justice at last.
The sentence included 10 years for manslaughter, five for causing the shipwreck, one for abandoning the ship before passengers and crew were clear, and one month for lying to the authorities afterwards.
Costa Crociere, the company that owned the Costa Concordia, sidestepped potential criminal charges in 2013 by agreeing to pay a €1 million ($1.1 million) fine.
Five of Francesco Schettino’s colleagues were also jailed for up to three years in earlier cases.
Costa Concordia cruise ship wreck will be removed from the coast of north-west Italy in June, officials say.
The stricken liner will then be taken away to be scrapped.
Ports in Italy, Britain, France, Turkey and China are bidding for the lucrative contract to dismantle the ship, Italian officials said.
The Costa Concordia hit a reef near the island of Giglio in January 2012 and capsized, killing 32 people.
Captain Francesco Schettino is currently on trial for multiple charges of manslaughter and for abandoning ship.
Officials overseeing the Costa Concordia salvage operation set out their timetable at a news conference on Friday.
Costa Concordia cruise ship wreck will be removed from the coast of north-west Italy in June
Project manager Franco Porcellacchia said that, from April, the team would start to fix at least 15 large tanks to the side of the ship. They will be filled with water, and then gradually emptied to give the ship enough buoyancy to float off the seabed.
Once afloat, the ship can then be towed away.
Franco Porcellacchia said if that does not work then the world’s largest semi-submergible ship, the Dockwise Vanguard, will be on standby and can literally carry the ship into port.
The winning contract for dismantling the Costa Concordia is due to be announced in March.
Italy’s environment minister Andrea Orlando said the preference was to keep the project in Italy, both to limit the environmental impact and to keep any economic benefits.
The 290m-long vessel was righted last September in one of the largest, most complex salvage operations ever that took 18 hours and followed months of stabilisation and preparation work by a team of 500 engineers and divers.
That operation allowed divers to retrieve the remains of one of the two people still missing in the disaster, believed to be an Italian passenger, Maria Grazia Trecarichi. An Indian waiter, Russel Rebello, is still unaccounted for.
Italian officials have revealed that divers scouring the site of the Costa Concordia wreck off the coast of Italy have recovered what could be more human remains.
It comes days after Italy’s civil protection agency said it had found remains that could belong to the last two missing victims of the disaster.
All the remains have been sent for DNA testing.
The cruise ship ran aground and partially sank off Giglio island last year with the loss of 32 lives.
Divers scouring the site of the Costa Concordia wreck off the coast of Italy have recovered what could be more human remains
It was raised upright last month in a major salvage operation.
“Other remains have also been found and are currently undergoing DNA tests,” the agency’s chief Franco Gabrielli told reporters on Wednesday.
“We are waiting for the results of the analysis,” he added.
Costa Concordia’s captain, Francesco Schettino, is on trial over the disaster.
Francesco Schettino is accused of manslaughter, causing the shipwreck and abandoning ship, but says he is being made a scapegoat for the errors of others.
Two people were reported missing, presumed dead, after the disaster – Indian waiter Russel Rebello and Italian passenger Maria Grazia Trecarichi.
It was thought they had been trapped beneath the ship and the rocks, correspondents say.
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