He also said a constitutional declaration aimed at curbing presidential powers had been cancelled.
Mohammed Mursi, who was elected in June, is a member of the Muslim Brotherhood.
Relations between the Brotherhood and the military have been tense since the fall of President Hosni Mubarak.
Field Marshal Mohamad Hussein Tantawi – who has also been removed as defence minister – has not yet indicated whether he accepts the moves.
Presidential spokesman Yasser Ali said a career army officer, General Abdel-Fattah el-Sissi, would replace Field Marshal Mohamad Hussein Tantawi.
The president’s intervention is clearly an attempt to take decisive action in his tug of war for control with the armed forces.
Under the interim constitutional declaration issued before Mohammed Mursi was sworn in, the president cannot rule on matters related to the military – including appointing its leaders.
The Supreme Council of the Armed Forces (SCAF), which issued the declaration, also dissolved parliament, which is dominated by the president’s Islamist allies.
As head of the SCAF, Field Marshal Mohamad Hussein Tantawi became Egypt’s interim ruler after President Mubarak was ousted following mass protests in February last year.
Sunday’s presidential announcement also said armed forces chief of staff Sami Annan was retiring.
The spokesman said Gen Annan and Field Marshal Mohamad Hussein Tantawi had been appointed as presidential advisers and were given Egypt’s highest state honor, the Grand Collar of the Nile.
Tensions between the Egyptian presidency and the military have been exacerbated since Islamist militants in the Sinai peninsula killed 16 border guards in a raid last week.
Both sides have tried to use the incident to strengthen their position.
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