The Coast Guard says the hull of the ship has not been compromised, and no injuries have been reported.
Officials say the 178-foot Spirit of Glacier Bay grounded just before 8 a.m. Monday.
The Coast Guard has deployed aircraft and ships to the site, and a National Park Service vessel from Glacier Bay also is en route.
The Spirit of Glacier Bay is home-ported in Juneau and owned by Cruise West Enterprises.
The Coast Guard says current weather conditions are clear skies and calm seas.