The voting will see half of Egypt's 51 million electoral heading to ballots. The next phase will take place on December 22.
The voting today is taking place in Cairo, Alexandria, Daqahliya, Gharbiya, Sharqiya, Assiut, Sohag, Aswan, North Sinai and South Sinai.
The run-up to the referendum saw deadly clashes between supports of the government and those against it.
The vote on the disputed charter is seen by many experts as a choice between Egypt moving closer toward a religious state led by Morsi's Muslim Brotherhood and the ultraconservative Salafi bloc, or one that retains the secular traditions with an Islamic character.
Statements from the country's High Elections Commission say that there are 7,000-10,000 judges supervising the polling.
This is the fifth elections Egyptians have since the January 25 revolution, the first being the referendum on March 2011 constitutional declaration, then the People's Assembly elections in November and December 2011, the Shura Council elections in January and February 2012 and the presidential elections in May and June 2012.
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Last evening, the National Salvation Front of opposition groups organised demonstrations at the presidential palace and in Cairo's iconic Tahrir Square, after deciding to call on followers to vote "no" in the referendum, rather than a boycott.
In Tahrir Square, protesters chanted, "Oh martyr we swear by your blood, another revolution anew," and "The people demand the fall of the regime". MORE