BBC
A French fishing vessel is taking the Costa Allegra to a nearby island where passengers will be disembarked before being transferred to the Seychelles.
A fire in the ship's generator room on Monday caused it to lose all power.
The ship is from the same fleet as the Costa Concordia, which capsized off the Italian coast in January, killing 32.
The BBC's Katy Watson on Seychelles says that the ship is being towed to the Desroches island, near Alphonse island, south-west of the Seychelles.
Once there the passengers will be disembarked to the one hotel on the island, and then await transfer to the main Seychelles island of Mahe.
Tugs sent from Mahe are due to meet up with the Costa Allegra at around 0900 GMT, our correspondent adds.
The vessel has no air-conditioning or cooking facilities and an emergency generator powering the radio "could fail at any minute", the Italian coast guard says.
The ship's owner, Costa Cruises, has stressed that the Allegra - which had been drifting about 32km (20 miles) from Alphonse Island, one of the atolls in the Seychelles - is steady and conditions are safe. No-one has been injured, officials said.
The company says it is liaising with passengers' families via their emergency contact numbers, and has reached two-thirds of them. Contact has been made with the relatives of all the crew.
Families in the UK can call the company's emergency line on 020 7940 3300.
'Ship stable'
Italian authorities have directed three merchant ships and another fishing vessel towards the stricken cruise ship.
Authorities in the Seychelles earlier said that a plane had flown overhead and reported that the ship did not appear to be in danger.
Costa Cruises said that the ship sent out a distress signal when the fire broke out, and all passengers and crew not involved in fighting the fire assembled at the muster stations.
Most electric lights on board the ship are off as the batteries are being used to keep essential machinery going.
A spokesman said the situation on board was calm and the ship was stable.
There are 636 passengers and 413 crew on board the Costa Allegra, which left Madagascar on Saturday.
It was due to arrive in the Seychelles on Tuesday.
Further destinations on its itinerary include Alexandria and Naples in the Mediterranean Sea.
Somali pirates are known to operate in the area where the Allegra was adrift, though they have never seized a cruise ship.
Nine members of the Italian navy's anti-pirate unit are on board the ship as a precaution.
A facility on Costa Cruises' website allowing people to track the Allegra's position says that "data transmission is temporarily suspended".
The company says the Allegra received its regularly scheduled maintenance in dry dock in October 2011.
The Costa Concordia ran aground off the Italian island of Giglio on 13 January.
The Concordia's captain, Francesco Schettino, has been accused of manslaughter, causing a shipwreck and abandoning ship before all those aboard were evacuated. He denies any wrongdoing
An Indian Navy aircraft took this picture of the ship following its distress call
An Italian cruise ship left adrift in the Indian Ocean with more than 1,000 people on board following a power failure, has been taken in tow.
A fire in the ship's generator room on Monday caused it to lose all power.
The ship is from the same fleet as the Costa Concordia, which capsized off the Italian coast in January, killing 32.
The BBC's Katy Watson on Seychelles says that the ship is being towed to the Desroches island, near Alphonse island, south-west of the Seychelles.
Once there the passengers will be disembarked to the one hotel on the island, and then await transfer to the main Seychelles island of Mahe.
Tugs sent from Mahe are due to meet up with the Costa Allegra at around 0900 GMT, our correspondent adds.
The vessel has no air-conditioning or cooking facilities and an emergency generator powering the radio "could fail at any minute", the Italian coast guard says.
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Nationalities of passengers
- 135 Italians
- 127 French
- 97 Austrians
- 90 Swiss
- 38 Germans
- 31 Britons
Earlier, Commander Cosimo Nicastro from the Italian coast guard confirmed that the French-registered ocean-going trawler had reached the ship.
The company says it is liaising with passengers' families via their emergency contact numbers, and has reached two-thirds of them. Contact has been made with the relatives of all the crew.
Families in the UK can call the company's emergency line on 020 7940 3300.
'Ship stable'
Italian authorities have directed three merchant ships and another fishing vessel towards the stricken cruise ship.
Authorities in the Seychelles earlier said that a plane had flown overhead and reported that the ship did not appear to be in danger.
Costa Cruises said that the ship sent out a distress signal when the fire broke out, and all passengers and crew not involved in fighting the fire assembled at the muster stations.
Most electric lights on board the ship are off as the batteries are being used to keep essential machinery going.
A spokesman said the situation on board was calm and the ship was stable.
It was due to arrive in the Seychelles on Tuesday.
Further destinations on its itinerary include Alexandria and Naples in the Mediterranean Sea.
Somali pirates are known to operate in the area where the Allegra was adrift, though they have never seized a cruise ship.
Nine members of the Italian navy's anti-pirate unit are on board the ship as a precaution.
A facility on Costa Cruises' website allowing people to track the Allegra's position says that "data transmission is temporarily suspended".
The company says the Allegra received its regularly scheduled maintenance in dry dock in October 2011.
The Costa Concordia ran aground off the Italian island of Giglio on 13 January.
The Concordia's captain, Francesco Schettino, has been accused of manslaughter, causing a shipwreck and abandoning ship before all those aboard were evacuated. He denies any wrongdoing
