Earlier this week, the 53-member AU bloc gave Mauritania's ruling military junta the October 6 deadline to "restore constitutional order" and to reinstate Abdallahi, the country's first democratically-elected president.
Abdallahi, who was removed from power in the west African country on August 6, has been under house arrest since the coup.
His prime minister, Yahya Ould Ahmed Waghf, was re-arrested on August 22 and placed under house arrest.
The ruling military have since been under pressure from the AU, the United Nations and the international community to reinstate Abdallahi, threatening sanctions and isolation.
A statement issued by the presidency of the state council on September 1 said a 22-minister government had been formed with effect from the previous day.
A number of Western powers including the US and France have refused to recognise the military government, denouncing it as "illegitimate".
In a rare press statement Saturday, Aziz admitted that the August coup was "abnormal".
"We are effectively in an abnormal situation in the absence of an elected president," he said.
But "you must recognise that all the other constitutional institutions (the National Assembly, the Senate) are working as usual," adding that he would hold a presidential election at some point in the future.