As the search entered its ninth day, rescuers still had no certainty of the whereabouts of the missing group, comprising a dozen boys aged 11-16 and their 25-year-old coach.
The Prime Minister's Office says Prime Minister Prayuth Chan-ocha "wishes to thank the tremendous efforts of all worldwide units that have come to assist the Thai authorities in rescuing the youth football team that was stuck in the caves in Chiang Rai".
Chiang Rai Governor Narongsak Osatanakorn said on Monday that the 13 were being rescued but warned they were not out of danger yet.
The Thai navy is already doing this short-term, sending teams with high-protein liquid food to feed the boys, keep them company and explore the cave infrastructure where they are to ensure it is safe.
It's unclear if the group has been removed from the cave just yet.
Twelve boys and their football coach trapped in a Thai cave have been found alive but will need to learn to dive or wait months for flooding to recede before they can get out, the army says.
"One of the rescuers says to them, in English, 'You are very strong".
Aisha Wiboonrungrueng, the mother of 11-year-old Chanin Wiboonrungrueng, smiled and hugged her family as news of their discovery spread.
He added: "We found them safe".
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It shows people jumping "into the shallow water in an attempt to rescue passengers in the burning boat", according to ABC News . He said both of his sisters, Stefanie and Brooke , were on the boat with their mother and stepfather Stacey and Paul Bender .
"I'm delighted. I don't know how to describe it in words.it's unimaginable", the father of one of the boys said when he heard the news.
Meanwhile, diving experts have taken food and rejuvenative mineral salts so the boys and coach can rebuild their strength after 10 days with nearly no food or fresh water.
"In theory, human beings can last 30 days (without food)", Mr Narongsak told reporters while the search was still ongoing, before the team was found.
The governor added that authorities are also trying to send food into the cave but were unsure if "they can eat it or not because it is nearly 10 days".
Thailand's prime minister on Monday evening thanked the worldwide experts and rescuers.
Efforts to locate the team in the network were complicated by rising water levels, which continued as the rain did not abate.
"If they drink the water in the caves and it makes them sick it could hasten the problem that they are in, but if they don't drink it then they are also in trouble", Anmar Mirza, the United States National Cave Rescue Commission coordinator, told AFP.
Thailand has been a nation transfixed by the plight of the "Wild Boar" team, with social media lighting up in support of the group and the country's deeply spiritual reflexes stirred into action.
Experts in cave rescues from around the world had gathered at the site. An official Australian group has now followed a United States military team, British cave experts, Chinese lifesaving responders and several other volunteer groups from various countries.