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The government says Mr Abdallahi is now free to travel and make speeches.
BBC: Mauritania frees deposed leader
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Mr Abdallahi gained most votes in the first round two weeks ago but failed to get the 50% needed for victory.
BBC: Sidi Ould Cheikh Abdallahi
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France and the US had cancelled their aid, pending Mr Abdallahi's release.
BBC: Mauritania frees deposed leader
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Mr Abdallahi became Mauritania's first democratically elected leader in 2007 after a coup two years earlier, partly instigated by General Mohamed Ould Abdelaziz.
BBC: Mauritania frees deposed leader
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He gained most votes in the capital, Nouakchott and his home Trarza region in the south-east, officials say, while Mr Abdallahi won Mauritania's other 11 regions.
BBC: Sidi Ould Cheikh Abdallahi
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His real name was Abdallahi Ould Hmeida, the agency said.
CNN: Algeria attack may have link to Libya camps
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On 6 August, Mr Abdallahi, as president, tried to dismiss four senior army officers, including Gen Abdelaziz, the head of the presidential guard, who responded by launching the coup.
BBC: Mauritania frees deposed leader
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Foreign governments, which supported Mr Abdallahi's government as a bulwark of stability against the perceived threat of Islamist extremists in the Sahel, the swathe of Africa just to the south of the Sahara desert, reacted angrily.
ECONOMIST: Mauritania