The tank, known as a sponson, was one of 19 being welded onto the sides of the stricken ship to help float it and tow it away as scheduled later this year.
The plan is to fill the tanks with water and then empty them to raise the 290-metre (951-foot) cruise ship.
The sponson, which had been fixed to the ship last month, was already under water but it bent some support structures as it collapsed, Italian media reported.
The Costa Concordia project is the biggest salvage operation of a passenger ship ever attempted.
The ship crashed into rocks off the Tuscan island of Giglio with 4,229 people on board and keeled over on the night of January 13, 2012 with the loss of 32 lives.
Its captain Francesco Schettino is on trial for manslaughter, causing a shipwreck and abandoning the vessel before all its passengers had been evacuated.