The charter was adopted by 64 per cent of voters in a two-stage referendum that ended last weekend, according to the electoral commission.
Only one in three of Egypt's 52 million voters cast a ballot, however, fuelling an opposition challenge to its legitimacy after fraud allegations were rejected.
The charter, and Morsi's tactics in steamrolling it through without consensus, generated weeks of protests, some of them bloody, such as ones on December 5 that degenerated into clashes that killed eight people and wounded hundreds.
Morsi, who promulgated the text overnight, was to give his speech to the nation at 6:00 pm (1600 GMT), his office said.
The United States, which gives $1.3 billion a year to Egypt's influential military, has called on Morsi to work to "bridge divisions" with the largely secular opposition.
"We have consistently supported the principle that democracy requires much more than simple majority rule," acting State Department spokesman Patrick Ventrell said in a statement on Tuesday.
"We hope all sides will re-commit themselves to condemn and prevent violence."
The opposition sees the new charter as possibly being used as a tool to introduce strict Islamic sharia law on the Arab world's most populous nation.
The head of the opposition National Salvation Front, Nobel peace prize laureate Mohamed ElBaradei tweeted that the constitution was "void" because it conflicts with international law in regard to "freedom of belief, expression, etc".
Morsi's Muslim Brotherhood, though, argues the constitution is an important first step to restoring stability to the country after nearly two years of instability following the revolution that toppled former leader Hosni Mubarak.
"Every country has a crisis. But crises end, and we are now on the right path," said Essam al-Erian, the deputy head of the Brotherhood's political arm, the Freedom and Justice Party.
The Brotherhood's leader, Mohamed Badie, tweeted late Tuesday that the text was "the constitution of revolutionary Egypt" and urged the nation to "start building our country's rebirth... men and women, Muslims and Christians".
Morsi's next move will be to call legislative elections due before the end of February.
Until then, the Islamist-dominated senate will tackle all legislative business, following the dissolution of the lower chamber by the constitutional court in June. On Wednesday, 90 new senators appointed by Morsi were admitted.
The opposition Front says it will vie for seats in the new parliament, which under the new charter is given broader powers.
The political crisis surrounding the charter has taken a heavy toll on Egypt's economy.
The state-owned Al-Ahram newspaper reported there was "Fear in the Egyptian street" after rating agency Standard and Poor's downgraded Egypt's long-term credit rating one notch to 'B-'.
It reported the government was now restricting travellers from leaving or entering the country with more than $10,000, a measure to apparently conserve dwindling foreign reserves.
The International Monetary Fund this month put on hold a crucial $4.8 billion loan to Egypt needed to stave off a looming unmanaged currency devaluation.
The social volatility was seen as making it difficult, or impossible, for Morsi to meet likely IMF demands for financial reforms, such as tax hikes and subsidy cuts. Earlier this month, he repealed tax increases that were to come into effect on alcohol, cigarettes, cement and steel.
Standard and Poor's on Monday knocked Egypt's long-term credit rating down one notch to 'B-', saying it saw political tensions remaining at an "elevated" level.
