Mr. Mandela was addressing the Council in his role as Facilitator of the Arusha peace process for Burundi, which he took on after the death of the previous mediator, former Tanzanian President, Julius Nyerere.
"Where even one single human being, one group of people, one nation, one part of the world labours under preventable suffering, it is the concern of all of us in a world drawn closer together than even before," he said.
Over the past year, and in the last few months in particular, there had been an intensification of violence, including attacks on the civilian population, Mr. Mandela added.
However, in spite of "grave difficulties" still ahead, he said, a lot of progress had been made since the start of negotiations. "In the past 18 months, the Arusha process has seen the establishment of four committees, each targeting particular aspects of the talks." Two of these committees -- one dealing with the nature of the conflict in Burundi and the issue of genocide, and the other with reconstruction and development -- had nearly finished their work.
The Council earlier heard from Secretary-General Kofi Annan, who said that "in no other country is it so easy to imagine a repetition of what we have all sworn must never be repeated: ethnic killing on a genocidal scale."
"I heartily welcome the involvement of President Mandela and place great hopes in his ability to revive the peace process," Mr. Annan added. "We here in the United Nations Secretariat are determined to give whatever help we can, and I am sure the Council will wish to do the same."
Since September alone, more than 300,000 men, women and children in the region surrounding the Burundi capital, Bujumbura, have been herded into camps as part of the government's policy of forcible relocation, the Secretary-General told the Council. "The humanitarian impact of this policy has been disastrous," he added.
At the end of its session the Security Council unanimously passed a resolution "warmly endorsing" Mr. Mandela as Facilitator and condemning continuing violence in Burundi "perpetrated by all parties."