Gen Abdul Fattah al-Sisi said the constitution had been suspended and the chief justice of the constitutional court would take on Mr Morsi’s powers.
He said Mr Morsi had “failed to meet the demands of the Egyptian people”.
Mr Morsi and at least two leaders of his Muslim Brotherhood’s political wing are being held in detention, Brotherhood officials say.
Anti-Morsi protesters in Cairo gave a huge cheer in response to Gen Sisi’s speech.
However, US President Barack Obama said he was “deeply concerned” by the latest turn of events and called for a swift return to civilian rule.
The army’s move to depose the president follows four days of mass street demonstrations against Mr Morsi – Egypt’s first freely elected president – and an ultimatum issued by the military which expired on Wednesday afternoon.
The military’s move against the Muslim Brotherhood was well planned. Intervention must have been in the mind of the army chief, Gen Sisi, for some time. Muslim Brotherhood activists were arrested. Islamist TV stations were taken off the air.
In the elections that followed the Arab uprisings of 2011, the Egyptian Muslim Brotherhood, and its offshoots across the region, did extremely well. That was because it was well organised, with a reputation for honesty, and took advantage of the failure of secular parties to organise themselves.
But now the Egyptian army has inflicted a serious blow on the Muslim Brotherhood. One question is whether the Brotherhood will be forced back underground, as it was when it was banned in the years before 2011.
The Brotherhood foreswore violence many years ago. But there are jihadist groups in Egypt that have not. The army has calculated it can handle any trouble. But it is playing for high stakes – the future of Egypt.
TV stations belonging to Mr Morsi’s Muslim Brotherhood went off air at the end of the speech and state news agency Mena said managers at the movement’s Misr25 channel had been arrested.
Security forces were also reported to have raided the Cairo offices of al-Jazeera’s Egyptian television channel Mubasher Misr, arresting several workers.
Clashes have erupted in the northern city of Marsa Matrouh, with at least four people killed.
Another person was reported killed in the coastal city of Alexandria. If confirmed, this would bring the death toll from the past 24 hours to 21.
A notice went up on Mr Morsi’s Facebook page denouncing the army move as a “military coup”.
The statement asked Egyptian citizens – both civilians and military – to “abide by the constitution and the law and not to respond to this coup”.
Two Muslim Brotherhood officials said separately that Mr Morsi had been detained by the Egyptian authorities, and that senior aide Essam el-Haddad was also among those held.
Earlier reports said security forces had imposed a travel ban on Mr Morsi and other leading figures in the Muslim Brotherhood.
‘Roadmap’ for the future
General Sisi said on state TV that the armed forces could not stay silent and blind to the call of the Egyptian masses.
He spoke of a new roadmap for the future, and said that the chief justice of the Supreme Constitutional Court, Adli Mansour, would be given the task of “running the country’s affairs during the transitional period until the election of a new president”.
After Gen Sisi’s address, both Pope Tawadros II – the head of the Coptic Church – and leading opposition figure Mohammed ElBaradei made short televised speeches about the new roadmap for Egypt’s future which they had agreed with the army.
Mr ElBaradei said the roadmap aimed for national reconciliation and represented a fresh start to the January 2011 revolution.
“This roadmap has been drafted by honourable people who seek the interests, first and foremost, of the country,” added Pope Tawadros.
Opposition leader and former Arab League chief Amr Moussa told AFP news agency that consultations for a government and reconciliation “will start from now”.
Reacting to the army announcement, President Obama said he hoped to see a quick restoration of a democratically elected government in Egypt.
He ordered a review of US foreign aid to Egypt which by US law must be suspended in the event an elected leader is deposed by a military coup.