The government argues that the draft constitution would, among other things, codify gender equality and ban female circumcision and underage marriage in the West African state.
But the proposal has sparked huge protests since October over fears that the real motive is to reset presidential term limits -- allowing Conde, 81, to run for a third spell in office later this year.
Speaking on national television, the president said on Friday it was "due to our national and regional responsibilities that we have accepted a slight postponement of the date of the elections".
"This is not a capitulation or a step backwards," he said, adding that "the people of Guinea will express their choice freely at the referendum".
While Conde did not publicly announce a date for the new vote, a letter from the leader to the West African bloc ECOWAS, seen by AFP, said the new poll should take place within two weeks.
The poll had been scheduled for Sunday alongside parliamentary elections -- also delayed in the poor but mineral-rich country of some 13 million people, which has a legacy of autocratic rule.
The long-running demonstrations over the constitution issue have sometimes turned violent, with at least 30 protesters and one gendarme killed to date.
Conde's announcement followed criticism of the electoral process from the African Union, European Union and The International Organisation of La Francophonie (OIF), which gathers French-speaking states.
The OIF said this week it had problems with around 2.5 million of the 7.7 million names on the electoral roll, pointing to duplicate registrations and people who had died.
The African Union also cancelled an electoral observation mission to Guinea on Friday, citing a "major controversy" with the roll.
Meanwhile the EU said in a statement that a "lack of inclusiveness and transparency casts doubt on the credibility of the upcoming elections".
Quiet streets
Sekou Conde, a cadre in the president's Rally of the Guinean People (RPG) party, said the vote had been postponed purely for technical reasons.
"It has nothing to do with the electoral roll," he said, adding that people had ransacked voting stations.
A Western diplomat, who declined to be named, said he thought the delay would make no difference anyway.
"This changes nothing," he said, adding that there would be no credible change to the problems with the electoral roll within two weeks.
The streets of the capital Conakry were quiet on Friday evening after the announcement, despite months of protests.
Ibrahima Diallo, the operations manager for the National Front for the Defence of the Constitution -- an alliance of opposition groups behind the protests -- said that demonstrations would continue until Conde shelved the referendum.
Although both the current constitution and the proposed new text limit presidential terms to two, critics fear that passing a new constitution would reset presidential term limits to zero.
This would potentially allow Conde to run again when his second term runs out at the end of the year.
Conde was a longtime opposition figure who became the nation's first democratically elected president in 2010 on promises to fight corruption. He was re-elected in 2015.
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