"Naturally this decision is not an easy one but the more I think about it the more I believe I can be of greater value to the Proteas as a fully focused batsman and senior player at this time or rebuilding our team," he told reporters on Wednesday.
"I am deeply grateful and privileged to have been the test captain and I wish to thank Cricket South Africa for the honour of captaining my country. It was enjoyable and indeed a great learning experience. I kind of felt someone else could do a better job."
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AB de Villiers will replace Amla for the rest of the series.
Amla scored a first-innings double century in the second Test, securing the draw for his team, but his form in recent months has been relatively poor.
"We respect Hashim's decision and the manner in which he thought about it and then communicated with me," Cricket South Africa chief executive officer Haroon Lorgat said.
"He still has a huge role to play in shaping the success of our team without the need for a leadership title. He is just that type of a person and we are very fortunate to have him in our stable."
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