Mr Morsi and his Islamist allies approved it in a rushed, all-night session without the participation of liberals and Christians.
Anger at Mr Morsi even spilled over into a mosque where the Islamist president joined weekly Friday prayers. In his sermon, the mosque's preacher compared Mr Morsi to Islam's Prophet Mohammed, saying the prophet had enjoyed vast powers as leader, giving a precedent for the same to happen now.
"No to tyranny!" congregants chanted, interrupting the cleric. Mr Morsi took to the podium and told the worshippers that he too objected to the language of the sheik and that one-man rule contradicts Islam.
Crowds of protesters marched from several locations in Cairo, converging in central Tahrir Square for what the opposition plans to be the second massive rally in a week against Mr Morsi. They chanted, "Constitution: Void!" and "The people want to bring down the regime."
The protests were sparked by the president's decrees a week ago granting himself sweeping powers and neutralising the judiciary, the last check on his authority.
The edicts tapped into a feeling among many Egyptians that Mr Morsi and the Muslim Brotherhood, from which he hails, are using their election victories to monopolise power and set up a new one-party state, nearly two years after the fall of autocrat Hosni Mubarak.
As a result, Egypt has been thrown into its most polarising and volatile crisis since Mr Mubarak's removal. In the past week, clashes between Mr Morsi's supporters and opponents left two dead and hundreds wounded and raised fears of further chaos. The Brotherhood and other Islamists plan their own massive rally backing Mr Morsi on Saturday.
The opposition must now decide how to deal with a nationwide referendum on the document, likely to come in mid-December: Boycott the vote to protest at what critics call a deeply flawed charter or try to use anger at Mr Morsi to rally the public to reject it in the referendum.
Egypt's top reform leader, Nobel Peace laureate Mohamed ElBaradei predicted the document "will go to the garbage bin of history." But Mr Morsi said the constitution's swift passage was necessary to get Egypt through a transitional period in which there has been no elected lower house of parliament. The courts dissolved the Brotherhood-led lower house elected last winter.