He said al-Bashir was in custody but declined to provide more details.
State television later broadcast footage of him taking the oath to become head of the council, alongside his new deputy, army chief of staff Lieutenant General Kamal Abdelmarouf.
He said the council was also imposing a 10pm (2000 GMT) to 4:00am (0200) GMT curfew.
"We do not reject a military council in principle, but we reject these people because they are from Bashir's regime", said Abdelhamid Ahmed, a 24-year-old doctor.
The Sudanese Professionals Association which has spearheaded the nationwide protest campaign hailed Ibn Ouf's departure as "a victory of the people's will".
"We have no ambition to hold the reins of power".
Sudan analyst and researcher Eric Reeves described the military's ouster of the longtime president as a "palace coup with al-Bashir as scapegoat".
American Airlines Cancels Boeing 737 MAX Flights Into June
Airlines are continuing to ground 737 Max planes, with American Airlines announcing an additional 5,000 cancellations on Monday. Indonesian airline Garuda is the only company so far to publicly disclose it has cancelled an order for the planes.
What followed was a series of protests by human activists groups.
At the compound, large tents were put up and people brought in food and handed out water as the crowd swelled, a Reuters witness said.
In Sudan, demonstrators defied the coup leaders' decision to stay in power for two years.
Defence minister Awad Ibn Ouf then announced the coup on state TV saying Sudan had been suffering from "poor management, corruption, and an absence of justice".
In the Darfur conflict, rebels among the territory's ethnic Central African community launched an insurgency in 2003, complaining of discrimination and oppression by the Arab-dominated Khartoum government. He had faced 16 weeks of demonstrations sparked by rising food costs, high unemployment and increasing repression during his three decades of autocratic rule.
"Ironically, the prospects for democratic transition may be more remote than when Bashir was in power as there's no centre of power with which to negotiate", said Alex de Waal, executive director of the World Peace Foundation at the Fletcher School of Law and Diplomacy at Tufts University. The US imposed sanctions on him in 2007 in relation to his alleged support for militia blamed for atrocities in Darfur.
The military council will be in place for a maximum of two years, it says, but could last only a month if the transition to civilian rule is managed smoothly.
Wreckage of 'crashed' Japanese stealth fighter jet found
F-35 Joint Programme Office, but there was now very little information available about the incident, the spokesperson said. The last time this happened the plane crashed near the Marine Corps Air Station Beaufort, South Carolina.
"No party will be excluded from the political process, including armed groups", he told the council during a meeting on Abyei, a contested border region claimed by Sudan and South Sudan.
The minister also announcedthe suspension of the Constitution, the dissolution of the National Assembly and a three-month state of emergency for three months. He also said there would be a curfew from 10 p.m.to 4 a.m.
In a challenge to the military council, several thousand protesters remained in front of the defense ministry compound, and in other parts of the capital, as the curfew went into effect.
Chanting "the regime has fallen", they poured into the open ground outside the army base.
"Sudan's military authorities should ensure that emergency laws are not used to undermine people's rights". 2019 in this still image taken from a video obtained from social media.
Bashir, who swept to power in a 1989 coup, was one of Africa's longest-serving presidents and is wanted by the International Criminal Court (ICC) on charges of genocide and war crimes.
Parliament OKs law barring Brexit without deal
Since then May has held talks with the Labour party in a bid to find a compromise solution that can win a majority in parliament. A party spokeswoman said: "We had further detailed and wide-ranging talks with Cabinet ministers and officials today".