The Jem rebels are thought to have crossed into Sudan from bases in Chad
Sudan says it has cut off diplomatic relations with Chad, blaming it for helping rebels from Darfur to launch an attack on Sudan's capital, Khartoum.
Both Chad and Jem rebels deny working together to launch the assault on the Khartoum suburb of Omdurman, which the rebels say they have taken control of.
The government said the rebel advance, the closest they have come to Khartoum, had been defeated.
An overnight curfew imposed on Khartoum has been lifted, but not in Omdurman.
Sudanese President Omar al-Bashir made the announcement that his country was breaking off diplomatic ties with Chad on state television.
"These forces [behind the Omdurman attack] are all basically Chadian forces supported and prepared by Chad and they moved from Chad under the leadership of [rebel chief] Khalil Ibrahim," President Bashir said in his televised statement.
He added his country reserved the right to retaliate against Chad.
Sudan has offered a reward of $125,000 (£64,000) for Mr Ibrahim's capture and information that leads to his arrest, Sudan TV reported.
But the Chairman of the Justice and Equality Movement (Jem) Legislative Council, Dr Eltahir Adam Elfaki, said the group acted alone.
"Jem is self-dependent. It is a force that built itself from the equipment of the Sudan government following our activities against the Sudan government," he told the BBC from Libya.
"We have defeated them in so many operations. And we are more than sufficient with equipment than to get if from Chad. Chad has nothing to do with this. It's only a Jem operation."
Chad's government said it regretted Sudan's "hasty decision" to break off diplomatic relations.
It also denied any involvement in the attack on Khartoum.
Chadian Information Minister Muhammad Hussein told the BBC his government had "nothing to do" with it and urged all opposition groups in Sudan "to join the Abuja agreement for peace".
The Sudanese authorities say they have defeated Jem, but are still looking for rebel fighters who escaped.
They say they do not want civilians to be caught in the crossfire as they continue to round up the insurgents.
On Saturday, Jem rebels claimed to have taken the Wadi Saidna air force base about 10 miles (16 km) north of Khartoum, the suburb of Omdurman across the River Nile from Khartoum, and to have entered the capital.
There is now an uneasy calm in Khartoum, although there are reports that fighting is continuing in the west of the city, says the BBC's Amber Henshaw in Khartoum.
Founded by Darfuri Muslims loyal to Islamist leader Hassan al-Turabi
Led by lawyer Khalil Ibrahim Muhammad
Believed to have up to 3,000 fighters
Now said to have fragmented into several disparate groups
Schools in the capital are closed and residents are being warned not to leave their homes until they hear it is safe to do so.
Residents in Omdurman said there had been some fighting on Sunday morning.
One witness said people were very scared, that the atmosphere was very tense. He said they were worried Jem fighters might attack their homes looking for shelter.
Sudanese state TV said Mr Ibrahim led Saturday's assault on Omdurman and had been injured and was now in hiding in the suburb.
But there has been no independent confirmation that Mr Ibrahim was in the city during the attack.
A rebel commander, Suleiman Sandal, told the BBC that the attack sent out a clear message that while there was no security in Darfur, there was none in Khartoum either.
Experts say Chad and Sudan are fighting a proxy war using each other's rebels to achieve their military objectives.
The Jem is one of several rebel groups fighting the government and pro-government Janjaweed militia in the western Darfur region since 2003 over alleged discrimination by the authorities in favour of Arabs.
The rebels have been involved in raids on government forces in the area before.
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