The military government threw out the old constitution when it took power in 2014, after months of political instability and sporadic violence in the country.
It says that if approved, the new constitution will be a major step towards returning to full democracy.
However, opponents call the vote unfair as campaigning has been restricted.
The Referendum Act, brought in to govern the referendum process, criminalizes “anyone who disseminates text, pictures or sounds that are inconsistent with the truth”.
Rights groups have said that the new law “restricts expression and access to information about the draft constitution”.
Offenders face a maximum sentence of 10 years in prison. Dozens of people have been charged in connection with the referendum.
Independent election observer groups have requested accreditation to monitor the vote, but this has not been granted by the Election Commission.
The 40 million voters will answer yes or no to the question: Do you accept the draft constitution? They will also be asked a supplementary question, whether or not the appointed senate should be allowed to join the lower house in selecting a prime minister.
If the majority of voters say yes, the draft becomes the constitution, enhancing the military government’s legitimacy in the run-up to an election which PM Prayuth Chan-ocha, who led the 2014 coup, has promised will happen next year.
If the constitution does not pass, what will happen is uncertain, but the military government will remain in control.
The junta argues that corrupt politicians are to blame for the last decade of instability and divisive politics.
Made public in March, the draft constitution proposes a voting system which would make it difficult for a single political party to win a majority of seats in the lower house.
One of the most controversial clauses calls for the 250-seat senate to be fully appointed by the military government.
Before the coup, just over half of the upper house seats were directly elected and the rest were appointed.
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