The San Jose was sunk in June 1708 off Colombia's Caribbean coast, during combat with British ships in the War of the Spanish Succession.
The galleon was the main ship in a treasure fleet carrying gold, silver and other valuable items to Spain from its American colonies.
Only a handful of the ship's crew of 600 survived when the San Jose sank.
A team of Colombian and foreign researchers studied winds and currents of the Caribbean 307 years ago and delved into archives in Spain and Colombia searching for clues.
Experts confirmed that they found the San Jose on November 27, Santos said.
The experts confirmed that they located the San Jose because the ship was carrying unique bronze cannons with engraved dolphins.
"The amount and type of the material leave no doubt of the identity" of the shipwreck, said Ernesto Montenegro, head of the Colombian Institute of Anthropology and History.
A US company, the Sea Search Armada, claimed to have earlier found the wreck, but the find was not confirmed and a legal dispute broke out with the government over ownership rights that was ultimately resolved in a US court in Colombia's favour.
There could be up to 1000 shipwrecks off the Caribbean coast of Colombia, but of those only between six and 10 had large treasure cargoes, anthropologist Fabian Sanabria said.
When the San Jose set sail from South America in 1708 she was carrying riches destined for King Philip V's war chest as he fought Britain in the War of Succession.
She never arrived. She was sunk after a 90-minute battle with HMS Expedition, part of a four-ship squadron hunting the Spanish treasure fleet in the southern Caribbean.
Commodore Charles Wager described the close quarters broadsides - and said the San Jose exploded with such intensity he could feel the heat from his own ship.
"I believe the ship's side blew out, for she caused a sea that came in our ports," he wrote. "She immediately sank with all her riches."
Some reports said she was carrying 116 chests of emeralds, two tons of platinum and the personal wealth of the Viceroy of Peru - making her the richest treasure ship ever lost in the western hemisphere.
The San Jose is estimated to be one of more than 1000 galleons and merchant vessels to have sunk among Colombia's coral reefs.
Earlier this year, treasure hunters said they had found US$4.5 million of Spanish gold coins off the coast of Florida, part of another of King Philip's treasure fleets shipwrecked in a hurricane while en route from Havana to Spain.
- Telegraph