Correa, a US-educated economist who has been in power since January, has staked his political career on the referendum, which has exposed sharp fault lines in Ecuador, saying he could resign if he fails to win a convincing victory.
Miriam Alvarez, a mother of three whose home was enveloped by molten lava from the Tungurahua volcano last year, said: "I am voting for the assembly because I believe in change.
Most of the country's 13.3 million inhabitants live in the Andean highlands.
More than half are mixed race and one-quarter are indigenous peoples.
Spanish is the official language, but many people speak indigenous Quechua and Jarvo languages.
The country is the world's top banana exporter, but oil is its leading shipment overseas.
It also exports coffee, cocoa, shrimps and fish products.
More than one-third of the workforce is employed in agriculture.
"This assembly will finally give the poor a voice."
More than half of the members of congress were fired last month after a dispute with Correa over the referendum.
They fought with police to get back into the chamber, but were sidelined when congress convened with substitutes taking their place.
Opposition politicians had been hostile to the referendum plan but Correa, a political outsider popular for promising to curb the influence of the traditional parties, finally won majority backing.
Some fear the referendum could allow Correa to become too powerful by centralising the government around himself.
But Ecuador's politics have been unstable, with eight presidents coming and going in a single decade, three of them toppled amid congressional unrest, and analysts say Ecuador's political landscape may change by the time of another election in September to select members of the assembly.
It is possible that popular Indian groups could temper their allegiance to Correa if he does not offer more radical policies, such as land reform and oil nationalisation.