The filling of the dam has been a source of tension between the Nile countries. Egypt and Sudan argue that Ethiopia has not provided sufficient guarantees to their water supply, which is highly dependent on the Nile River.
"As (Ramaphosa) is a good friend for both Ethiopia and Egypt and also as incoming AU chair, he can make a discussion between both parties to solve the issue peacefully," Abiy said at a press conference in the South African capital Pretoria.
Egypt fears its water supply could be threatened by the dam
'A solution can be found'
For his part, Ramaphosa said he had accepted the task and that he had already reached out to Egyptian President Abdel Fattah el-Sissi.
"The Nile River is important to both countries and there must be a way in which both their interests can be addressed," said Ramaphosa. "There must be a way in which a solution can be found."
Concerns over the Renaissance Dam on the Blue Nile, one of the main sources of the Nile River, have dogged relations between the African nations for years. At times, Egyptian officials have threatened military action against the dam, including airstrikes, saying its existence poses an existential risk to Egypt.
For Ethiopia though, the dam is a much-needed source of power to energize what has become one of Africa's fastest growing economies.
Ethiopia and South Africa also signed several trade agreements spanning health, tourism and telecommunications industries during Abiy's visit.