Franco Gabrielli, from Italy’s Civil Protection Department, said it was a "great sorrow" that the final body had not yet been found, and pledged that divers will continue to search for Rebello.
Gabrielli told reporters that while the weather was not conducive for the mammoth task of extracting the wreck, it was good enough to allow it to start.
The giant liner’s wreckage was floating by midday on Monday, and was due to depart Giglio on July 21 on its five-day journey to Genoa, where it will used for scrap.
Rebello’s family have set up a Facebook page to celebrate his life. In a post, Rebello’s brother, Kevin Rebello, said he was watching the latest salvage operation live online, since he could not make it to Giglio.
He posted: "I strongly believe that they will find the body of my dear brother."