The ship was supposed to disembark in Punta Arenas, Chile, but those plans changed.
The guests have not been checked for the coronavirus since there are no tests available on the ship.
"Since it is flu season, and COVID-19 is not available onboard, it is difficult to determine the cause of these elevated cases at this time," a company statement said, according to CNN.
There are 1,243 guests on the ship and all have been told to stay in their cabins. Meals are being delivered by room service, the South Florida Sun-Sentinel reported.
They are also being provided free internet and telephone service to stay in touch with family on land.
Of the 586 crew members, those who are not essential to the ship operations are in quarantine. Those who are running the vessel self-isolate when not on duty.
The Zaandam embarked on a trip on March 7 from Buenos Aires, Argentina, with the voyage scheduled to end in San Antonio, Chile, on Saturday, March 21. The company announced its intention to suspend operations on March 17 as the Zaandam was still on its sailing.
Passengers have not left the ship since March 14, the Sun-Sentinel reported.
It had been anchored for several days in Valparaiso, Chile, while it was provisioned and refueled.
The ship still has to get permission to sail through the Panama Canal en route to Port Everglades in Southern Florida with an expected arrival date of Monday, March 30, but company officials are coming up with a contingency plan if the canal passage is denied, CNN reported.
Holland America's Maarsdam is also looking for a port to allow passengers off the ship, the Sun-Sentinel reported.
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