Authorities and relatives of the miners hugged, climbed a nearby hill, planted 33 flags and sang the national anthem Sunday after a probe sent some 2,257 feet (688 meters) deep into the mine came back with the note. "Today all of Chile is crying with excitement and joy," President Sebastian Pinera said.
The miners' ordeal may have just begun: Rescuers say it could take four months - until around Christmas - to get them out.
The men already have been trapped underground longer than all but a few miners in recent history. In 1983, two coal miners rescued after 23 days in northeast China; few other rescues have taken more than two weeks.
The miners have survived by drinking underground water, reports CBS News correspondent John Bentley. Now rescuers will be able to send down food, water, and supplies until they can reach them.
For the moment, however, news that the men even survived an Aug. 5 tunnel collapse outshines all other details.
"I'm happy, happy," said one of miner Mario Gomez's three daughters. "For the first time, I'll be able to sleep peacefully."
Pinera told Cooperativa radio that he saw video of the miners thanks to a camera sent through the probe shaft.
"I saw eight or nine of them. They were waving their hands. They got close to the camera and we could see their eyes, their joy," Pinera said.
Word of the miners' survival was a rush of good news in a country still rebuilding from a magnitude 8.8 earthquake Feb. 27 and its resulting tsunami, which together killed at least 521 people and left 200,000 homeless.
Mine officials and relatives of the workers had hoped the men reached a shelter inside the mine when the tunnel collapsed the San Jose gold and copper mine about 530 miles (850 kilometers) north of the capital, Santiago. But they had said air and food supplies would only last 48 hours.
Rescuers drilled repeatedly in an effort to reach the shelter, but failed seven times; they blamed the errors on the mining company's maps. Hopes rose after the eighth attempt early Sunday when rescuers heard hammering sounds.
Crews sent down a probe, then pulled it up with two notes the trapped miners had placed inside, including the one Pinera read. Gomez, 63, wrote the other note to his wife, confirming the miners' location underground and saying he loved her.
"When the mining minister said he had sent me a note, I couldn't believe it," said Gomez's wife, Lila Ramirez. "I know my husband is strong, and at 63, is the most experienced miner who could lead his co-workers. But no more mining (for him.)"
Gomez wrote that the miners used vehicles for light and a backhoe to dig a canal to retrieve underground water.
The opening the miners used to deliver the notes is not wide enough to haul up the miners. Rescue equipment brought from outside the country was being assembled Sunday to dig a tunnel 27 inches (68 centimeters) in diameter through which the miners will eventually be brought to the surface.
The hole already drilled will be used to send down small capsules containing food, water and oxygen if necessary, and sound and video equipment so the miners can better communicate with loved ones and rescuers.
This spring 115 Chinese miners were rescued after being trapped for more than a week because workers digging tunnels had broken into a water-filled abandoned shaft. The accident killed 38 miners.
The Chilean mine was closed temporarily for safety reasons in 2005.
Chile's drama months after the earthquake shares some parallels with the 2002 Quecreek Mine accident, in which nine Pennsylvania miners were trapped in a flooded tunnel about 15 miles from where one of the planes hijacked on Sept. 11 crashed some 10 months earlier. Americans cheered as all nine miners were rescued three days later.
Hundreds of workers are using equipment from the United States and Australia in the Chilean rescue.
Both the company that owns the mine, San Esteban, and the National Mining and Geology Service have been criticized for allegedly failing to comply with regulations. In 2007, an explosion at the San Jose mine killed three workers.
Chile is the world's top copper producer and a leading gold producer.