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The investigation will be led by Afghan defence minister Abdul Rahim Wardak, seen as close to Karzai, a statement from the presidential palace said.
The statement added: "As a result of actions taken to identify the elements behind Rabbani's assassination, an arrest has been made and information obtained from him is bringing us closer to the truth."
A government source speaking on condition of anonymity told AFP that the arrested man had spoken to the bomber the day before the attack.
Copyright AFP (Agence France-Presse), 2011