Rescuers are now considering how best to bring the group to safety.
Engineers have been pumping water out of the cave for a week and a temporary respite from the rains has lowered water levels.
"We haven't eaten", a boy said in Thai, then in English: "We have to eat, eat, eat!"
Now that the 12 boys and the soccer coach have been found inside the sprawling Tham Luang cave complex, all attention is now on how rescuers can get them out.
As they wait, the BBC's Howard Johnson reports, more than 2,500 gallons of water per hour are being pumped out of the cave in order to ease the boys' dive out.
An underground headquarters has been set up, stocked with dive gear, oxygen tanks, medical supplies and food, which will be transported to the boys and their coach.
Thai authorities are committed to "100 percent safety" in extracting the boys and their coach from a partially flooded network of caves, Chiang Rai Governor Narongsak Osatanakorn said Tuesday morning, according to the Associated Press. "They may try to bring them out at night".
The governor said that requests had been made to build "infrastructure" in the cave leading to the pocket where the teenagers and their coach are located.
Rear Admiral Apagorn Youkonggaew, head of the Thai navy's special forces, told reporters there was "no need to worry". In footage that emerged after the boys were found by two British divers late Monday one asks to "go outside". They went missing on June 23 while exploring the six-mile-long cave, which is in a park in northern Thailand near the Myanmar border.
John Volanthen and Richard "Rick" Stanton are expert cave divers skilled in navigating perilous underwater passages, a colleague says - and they've been called overseas before to help with hard rescues. Thirteen ambulances are on standby to transport them.
The search for the group had gripped the nation as it was unclear where they were or whether they were even still alive.
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Family members of the missing hugged each other as they cheered.
The Thai prime minister, Prayuth Chan-ocha, held a telephone conference to congratulate the rescuers at the cave site.
"Thank you all Thais, thank you all foreigners", he said. But monsoon rains flooded several chambers and blocked their exit. It was most likely dry when they entered, but rushing waters later clogged the way back with mud and debris.
Narongsak said the team was found between 300 and 400 metres past a section of the cave that was on higher ground and was thought to be where the team members and their coach may have taken shelter.
Except for narrow holes, two divers would escort each of the boys.
But if that is the avenue rescuers intend to take, they will first need to wait for the boys to regain their strength.
British cave-diver John Volanthen walks out from Tham Luang Nang Non cave in full kit.
The matter is complicated, he added, by the fact the boys don't know how to swim. That could take as long as months, however, given that Thailand's rainy season typically lasts through October. That was despite the team having capabilities of working in zero visibility areas, land search and rescue, and diving in flooded caves.
Relatives of the missing children share photos of them after the 12 boys and their soccer coach have been found alive in a cave They'd been missing for over a week.
Explorers have spent days scouring the mountaintop for possible alternative openings. "We have taken care of those boys following the doctor's recommendation".
Although they have had no shortage of water, one of the boys is heard to say: "Eat, eat, eat, tell them we are hungry". "I want to tell him I'm still here waiting", Kieng Khamleu, said of her son Pornchai Khamleung inside the cave.