The Costa Allegra cruise ship, operated by Costa Cruises, is now being towed to safety after a fire onboard the liner caused the ship to lose power.
The vessel, owned by same company which operated the ill-fated Costa Concordia, was cast adrift in the Indian Ocean and left at the mercy of the current as it waited for tugs to arrive to tow it to a port. It is expected to arrive on the island of Desroches in around an hour's time.
More than 30 Britons were among the 636 passengers and 413 crew members stranded on the Costa Allegra, after a blaze in an electric generator room.
A French vessel reached an Italian cruise ship overnight, the Seychelles coastguard said.
The authorities were making arrangements for the evacuation of those on board to the island - and then their transfer to Seychelles' main island.
Somali pirates are active in the region but have not been sighted near the ship.
Among the holidaymakers on board were 31 Britons and one Irish person. The majority were Italian, French, Austrian and Swiss.
The incident came after the Costa Concordia cruise liner, also operated by Costa Cruises, struck rocks off the west coast of Italy on January 13, leaving a death toll expected to reach 32.
The 28,597 gross tonne ship had armed security on board and fears that it could be targeted by Somali pirates were played down.
The ship was sailing around 200 miles off the south west of the Seychelles when the fire broke out.
Officials said it had been extinguished without causing any injuries or casualties and the liner had remained stable and upright despite 5ft waves battering it.
Captain Giorgio Moretti said the air conditioning and lights had been knocked out when the power went, but officials said emergency generators were keeping the ship's command room lit and instruments such as its radio functioning.
The rest of the ship was apparently dark, however, as night fell and its passengers were being kept in the large communal rooms rather than their cabins.
Seychelles presidential spokeswoman Srdjana Janosevic said: "If pirates attack, the armed guards on board will respond.
"But as far as I am aware, no pirates have been sighted in the area."
The Allegra left northern Madagascar on Saturday and was cruising towards the port of Victoria, Seychelles' capital, when the blaze broke out.
It had been due in Victoria today.
Costa Cruises said in a statement earlier: "The shipboard fire-extinguishing system and procedures were promptly activated and the special firefighting squads intervened to extinguish (the) fire."
The company said the general emergency alarm had been sounded "as a precaution" and that passengers had gone to the muster stations.
"According to the procedures, the relevant authorities were alerted, including the Maritime Rescue Control Centre in Rome, which are following the situation ready to provide the necessary support," the statement said.
The Foreign Office said it was aware of the incident but could not confirm the number of Britons on board the ship or say whether it was providing consular assistance.
Built in Genoa in Italy in 1992, the vessel has eight passenger decks and can carry a maximum of 1,400 people, including crew.
Costa Cruises said it was keeping in touch with guests' families via their emergency contact numbers and was also communicating with the families of crew - most of whom were Italian and eastern European - through local crew-recruiting organisations.
The statement said: "Costa Allegra is in compliance with all certificates required by both national and international regulations. Moreover, in October 2011 the ship received its regularly scheduled maintenance in dry dock."