Thai officials gave food and water to hundreds of emaciated Rohingya who have been at sea on the overcrowded boat for weeks. In recent days they were abandoned by their traffickers, who also disabled the engine before fleeing.
"We fixed the engine and the boat left last night after 3:00 am," Satun provincial governor, Dejrat Limsiri, told AFP, in line with Thailand's policy of "helping on" such vessels.
"We gave them ready-to-eat meals. They are now out of Thailand territory... they will try to go to Indonesia as it seems they cannot get to Malaysia," he said.
"They are heading along the route to Aceh," he added, referring to the Indonesian province where another migrant boat landed at the weekend.
The boat that had been in Thai waters, with around 300 people on board including many young children, was found drifting on Thursday. Its bedraggled passengers were visibly emaciated and pleaded for help.
In dramatic scenes later in the day, a Thai navy helicopter dropped food packages into the water, prompting several men to dive into the sea to retrieve them -- some eating raw instant noodles as they swam back.
Around 10 people had died on the boat during the last few days, passengers told reporters who drew up alongside.
The Satun governor said authorities had helped the migrants on towards their intended destination further south, but denied pushing the boat out of Thai waters.
"They want to go to Malaysia but they cannot go now. They do not want to come to Thailand as they know they will face legal action... and will be sent back to Myanmar," he said.
The International Organization for Migration (IOM), which has called on governments to allow sea-stranded migrants to disembark, confirmed that the migrants refused to land in Thailand.
"The Thais agreed to allow them to disembark but they said they want to carry on... to go to Malaysia," Bangkok-based IOM spokesman Joe Lowry told AFP.
"If they don't want to disembark you can't force them to do it," he added.