Costa Allegra was towed into the islands’ main port, Victoria, by a French fishing boat.
More than 1,000 passengers are on board and the evacuation is expected to take several hours.
Costa Allegra, which lost power on Monday, is from the same fleet as the Costa Concordia, which capsized off the Italian coast in January, killing 32.
As the Costa Allegra has no power, bringing the ship alongside the dock so that passengers can be disembarked is expected to take at least two hours.
Passengers who are in the weakest medical condition will be brought off first.
Those on board have spent three days without power or hot food, with supplies flown in by helicopter.
The vessel has also had no functioning air conditioning. Most of the passengers stayed on deck to benefit from the cooling breeze in the hot weather.
Accommodation is being arranged, but with hotels already busy, most are expected to fly straight to Italy.
Planes are being lined up in the Seychelles to fly home passengers from the vessel as soon as possible.
Costa Cruises, the company which owns the Costa Allegra, says it received its regularly scheduled maintenance in dry dock in October 2011.
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