The ship, called the San Jose, was part of the fleet of King Philip V as he fought the English during the War of Spanish Succession.
The galleon sank in the Caribbean Sea close to the walled port city of Cartagena and it is speculated to be carrying 11 million gold coins and the bodies of 600 people.
A team of international experts, the Colombian navy and the country's archaeology institute discovered the wreck last week near the island of Baru, Colombia's president, Juan Manuel Santos, said.
In a message posted on Twitter, the president conveyed his excitement at the find.
'Great news! We have found the San Jose galleon. Tomorrow we will provide details at a press conference from Cartagena,' the message read.
Efforts to retrieve the sunken treasure were documented in a video posted on the government's website.
In a broadcast to the nation, Mr Santos said the discovery is the largest haul of national treasure ever lost at sea.
"It constitutes one of the greatest - if not the biggest, as some say - discoveries of submerged patrimony in the history of mankind," he said.
"The Colombian government will continue its investigative process of exploring, of protecting submerged cultural patrimony."
He said a museum would be built in Cartagena to house the artefacts.
"It's a scientific event that reminds us that Colombian history is constituted by events of very different eras protagonised by thousands of people who make up our national memory," he said.
The government's claim on Friday did not shed light on a legal wrangle with Sea Search Armada, a US-based salvage company which had a long-standing suit against Bogota over ownership of the wreck. SSA said in 1981 it had located the area in which the ship sank.
SSA and the government were partners back then and following international custom, they agreed to split any proceeds. The government later said any treasure would belong to Colombia.
In 2011 a US court declared the galleon property of the Colombian state.
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