The search and rescue operation has involved dozens of global experts and rescuers, including a US military team.
Jedsada Chokdumrongsuk, permanent secretary at the Public Health Ministry, said the first four boys rescued, aged 12 to 16, are now able to eat normal food, though they can not yet take the spicy dishes favoured by many Thais.
Mr Nargonsak said this phase may take longer than the previous two rescue missions.
Extracting the whole group could take three to four days and depended on the weather, an army commander involved in the mission said. They have received training in recent days in preparation for the extraction effort, but they will have to swim using scuba gear through fast-flowing water in darkness, a challenge for even elite divers.
One of soccer's most popular teams, Manchester United, expressed its relief over the rescue and invited the boys and their coach, as well as those who saved them, to come see them play on their home ground this season.
The group had been missing for nine days before they were found huddled inside the cave and survived by drinking water dripping off the stalactites above them.
Rescuers have suggested that it may take up to two days for the rescue to be completed, with each boy being brought out individually.
But rescuers were also waging a "war with water and time" before expected heavy rains.
The plight of 12 Thai boys and their coach from the "Wild Boar" football team has transfixed Thailand since they became trapped in a cramped chamber several kilometers inside the Tham Luang cave complex on June 23.
Following the evacuation announcement, a rescue operation was launched and the first boys could emerge from the cave as early as 9pm local time (2pm GMT).
England fans go wild at Odsal
England were comfortable winners over Sweden in their last game, winning 2-0 thanks to goals from Harry Maguire and Dele Alli. France has been one of the most complete sides in the tournament , topping Belgium in the semifinal round.
British divers Richard Stanton and John Volanthen discovered the boys on Monday.
A Thai governor says the operation to bring out 12 schoolboys and their soccer coach from deep inside a cave where they have been trapped for two weeks has begun. Monsoon flooding cut off their escape and prevented rescuers from finding them for nearly 10 days. "Extremely robust", he said, adding it would take eight hours to build and 17 hours to send to Thailand. Much of the path out of the cave is flooded, and in places the cave gets as narrow as 70cm.
Musk on Tuesday visited the cave and posted pictures and videos online.
President Donald Trump tweeted Sunday: "The U.S.is working very closely with the Government of Thailand to help get all of the children out of the cave and to safety".
The boys sounded calm and reassuring in handwritten notes to their families that were made public Saturday.
Another, identified as Tun, writes: "Mom and Dad, please don't worry, I am fine".
"Don't be anxious", wrote another boy, Mick.
One boy named Mick wrote: "Don't be anxious, I miss everyone". "To the parents of all the kids, right now the kids are all fine, the crew are taking good care". I promise I will care for them the best I can. "I suspect the Thai govt has this under control, but am happy to help if there is a way to do so", he wrote on Twitter.
Another letter read, "I'm fine but it's a little bit cold, don't worry and don't forget my birthday". "I want to say thanks for all the support and I want to apologize to the parents", he wrote.
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