Four tugs were used to try and pull the Mona Lisa free from a sandbank and a number of other measures also proved unsuccessful prompting a decision to evacuate the ship's 657 passengers.
The passengers, mainly elderly Germans, are to be ferried to the port of Ventspils by Latvian naval vessels. There are no reports of any casualties.
The Mona Lisa is registered in the Bahamas. It has a crew of 327 and was on route from Poland's Gdansk to Latvia's Riga, when it ran aground 18 km (11 miles) off the coast of Latvia.
Latvian Defense Minister Vinets Veldre visited the Mona Lisa last night and spoke to the captain of the liner, who according to the minister was to blame for the incident.
"The cause of the incident could be the captain's decision to deviate from the set course so that passengers could take pictures against the background of the Irben lighthouse," the minister said.
The Mona Lisa is 173 meters (567.5 feet) in length. It was constructed in 1966 and it has ran aground nine times since then.