The immediate beneficiaries are Benin, Burkina Faso, Ethiopia, Ghana, Madagascar, Mali, Mozambique, Niger, Rwanda, Senegal, Tanzania, Uganda and Zambia, the bank said in a statement.
Donald Kaberuka, the bank's president, said on Wednesday that "this is a significant step forward in the initiatives by donor countries, both Group of Eight and others, on the issue of debt, which is to be applauded highly".
The bank said the debt relief was intended to allow the 13 countries to channel resources into poverty relief, but also to stimulate economic growth and encourage governments to stay on track with reforms.
It urged the countries to fight corruption.
The G8 powers in July announced their plan to cancel the debts of 18 of the poorest countries owed to the World Bank, the IMF and the African Development Bank.
On the basis of country eligibility assessments independently conducted by the IMF, International Development Association (IDA) and African Development Fund (ADF), 13 countries became the immediate beneficiaries of ADF debt relief.
Mauritania will qualify for debt relief if it makes reforms.
The African Development Bank Group, established in 1964, is a multinational development bank supported by 77 nations from Africa, North and South America, Europe and Asia. Its headquarters are in Abidjan, Ivory Coast.