The proposed constitution says Islamic law will not be applied in the mainly Christian and animist south and removes a requirement that the president be a Muslim.
"Sudan is an all-embracing homeland where religions and cultures are sources of strength, harmony and inspiration," the draft declares.
Debate is expected to take a week. A vote is expected shortly before a government of national unity takes office on 9 July, with Omar al-Bashir continuing as president and Ali Osman Mohammed Taha - the lead negotiator in peace talks with the southern rebels - and former rebel John Garang as vice-presidents.
Lawmakers are expected to unanimously adopt the constitution, as they did with the peace deal struck in January that ended a two-decade war between government troops and Garang's forces.
When the constitution was presented to al-Bashir at a ceremony on Sunday, cheers of "God is Great" and "Hallelujah" broke out and a band played a fanfare on trumpets and clarinets.
"This is the really the beginning of a democratic change in the country"
Ismail Hajj Mussa, head of parliament's legal and constitutional committee
"This is a message we send to the whole world from the Sudanese people: This is a tolerant people," al-Bashir said on Sunday. "We are a people who can overcome all suffering and, sitting together face-to-face, we can forget all our suffering."
Mahjoub Mohamed Salih, the 75-year-old editor of the oldest independent newspaper now in circulation, Al-Ayam, wrote on Monday that profound change was ahead for Sudan, starting with the 9 July convening of the unity government.
"There should be changes in the life of people, the situation should be normalised and the doors should be wide open for political activities, and all restrictions should be lifted and there should as well be freedom of expression and of organisations," he wrote.
Ismail Hajj Mussa, who heads parliament's legal and constitutional committee, said the current constitution, adopted in 1998, was the work of the ruling party and the peace agreement the work of the ruling party and the rebels.
The proposed constitution, though, was the work of "a host" of groups, ironed out during more than 300 hours of meetings and discussions held around Sudan.
Sudanese officials in talks with
"This is why it is well received," Mussa said on Monday. "I believe this is a qualitative development in the constitutional life [of Sudan]," he added.
"This is the really the beginning of a democratic change in the country," he added.
The draft constitution said Islamic law would be the source of legislation applied in the northern states, while in the south local customs and traditions and respect for the country's diversity would inform laws.
The president is no longer has to be a Muslim, but must be Sudanese, at least 40 years old, literate and free of a criminal record.
It is the first Sudanese charter to lay out freedoms of religion and expression as human rights. In another first, women as well as men would be able to pass on Sudanese nationality to their children.
The rebels took up arms demanding more autonomy for the south in 1983, the same year a series of Islamic laws were enacted by Khartoum.
The northern-based government - under pressure from the US and spurred by suspicion of militant Islam following the September 11 attacks on the US and weary of international political and economic isolation - had begun to move away from Islamic law even before the January peace deal that paved the way for the constitution.
Although the civil war was fuelled by historical disputes and competition for oil and other resources, religion played a role and was an issue during the peace talks.
The talks stalled at one point over rebel insistence that the capital, Khartoum, should not fall under Islamic law.
The rebels eventually yielded on that point, with the understanding that non-Muslims would not be harassed - as they allegedly have been in the past - in Khartoum.