Rescuers are trying to reach a third survivor, hours after a honeymooning South Korean couple were rescued.
Three people are so far confirmed dead and about 40 are unaccounted for.
First officer Ciro Ambrosio has also been detained.
The ship’s operator, Costa Crociere, said the vessel had been following its regular course when it hit a submerged rock.
In a TV interview, ship’s captain Francesco Schettino said the rock was not marked on any maritime charts.
Police are investigating why the accident happened in calm conditions.
The South Korean couple was located after rescuers heard voices from a cabin two decks down on the half-submerged ship late on Saturday. They were reached a few hours later.
The newlyweds, both 29 years old, were both in good health when they were brought ashore.
Later on Sunday morning, rescuers located and spoke to the third person found inside the ship. Officials say he is a senior Italian member of the crew, and they believe he has a serious injury to his leg.
Two scuba divers were smashing a window on the ship trying to get to someone inside.
Two French passengers and a Peruvian crew member died, and another 30 people were injured, two seriously.
Divers are continuing to search the ship, which is lying on its side near the Tuscan island of Giglio.
President of Costa Cruises, Gianni Onorato, said the main task for the company was now to assist survivors and help repatriate them.
He said it was difficult to determine what had happened, but that the ship had experienced a blackout after hitting “a big rock”.
Gianni Onorato added: “We will be working in full transparency with Italian authorities to understand the causes of the disaster.”
He said normal lifeboat evacuation had become “almost impossible” because the ship had listed so quickly.
Francesco Schettino, the 52-year-old captain, had worked for Costa Cruises for 11 years.
The chief prosecutor in the city of Grosseto told reporters that Captain Francesco Schettino “very ineptly got close to Giglio”, according to Italy’s Ansa news agency.
Coast guard captain Cosimo Nicastro told Italian TV that divers had carried out an extensive search of the waters near the vessel and found no further bodies.
On Saturday, survivors were taken to Porta San Stefano, about 25km (15 miles) away on the mainland.
Many arrived there still wrapped in blankets, and some were clearly very shaken by what they had endured.
Passenger Luciano Castro told Ansa news agency: “We heard a loud noise while we were at dinner as if the keel of the ship hit something.”
“The ship started taking in water through the hole and began tilting.”
Some passengers told the Associated Press news agency that the crew had failed to give instructions on how to evacuate the ship.
Several passengers compared the accident to the film Titanic, about the sinking of the giant ocean liner in April 1912 which claimed more than 1,500 lives.
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