The person who tested positive was "provided private air transportation back to the comforts of home, ensuring that our safety measures were maintained from door-to-door," the statement said.
Cruise line spokesperson Susan Lomax said the ship was in Skagway on Tuesday and would be at sea before returning to Seattle on Friday.
Robert Barr, Juneau's deputy city manager, told KTOO Public Media that the person had been aboard the Celebrity Millennium, which stopped in Alaska's capital city on Monday.
"I think that we were never expecting the cruise season to be entirely COVID free," Barr told the outlet. He said he was confident in the city's efforts with the cruise industry and state health department to ensure a safe season.