The positive individuals, who were sharing a room onboard, were discovered while the ship was conducting mandatory end-of-cruise testing, completed 72 hours before disembarkation.
Royal Caribbean, which owns the Celebrity Cruises brand, said in a statement that the guests were “asymptomatic and currently in isolation and being monitored by our medical team”.
It added: “We are conducting contact tracing, expediting testing for all close contacts and closely monitoring the situation.”
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According to Cruise Mapper, the ship left St Maarten on Saturday, before sailing to Barbados and Aruba before docking in Curacao. It has now left the island, and is heading back towards St Maarten.
Everybody onboard Celebrity Millennium had been fully vaccinated, including all crew and guests.
All guests were required to show proof of vaccination as well as a negative Covid test taken within 72 hours of departure from St Maarten, as required by Royal Caribbean.
The cruise industry worldwide has been decimated by the Covid crisis, following high-profile outbreaks at the beginning of last year.
US cruise lines are gearing up to restart operations this summer, while rules for UK cruising mandate that ships must only accept those resident in Britain or the Common Travel Area, and passenger numbers be capped at 1,000.
Some cruise lines, including Royal Caribbean, Crystal Cruises and Saga, require that all passengers be fully vaccinated before boarding.
Watch: Will the full easing of restrictions in England go ahead on June 21?