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DUBLIN, Ireland (CNN) -- Irish Prime Minister Bertie Ahern issued an ultimatum Tuesday to republican paramilitary groups which have yet to declare cease-fires, ordering them to do so within 24 hours.

One such group, the Real IRA, a dissident faction of the Irish Republican Army, claimed responsibility for the car bomb in the Northern Irish town of Omagh that killed 28 people and injured 220 on Saturday.

The Real IRA, using a recognized codeword, made the claim in a phone call to an Irish newspaper on Tuesday.

A male caller apologized for the death toll, saying the group had given three warnings before the explosion and had not been aiming to kill civilians.

"Despite media reports, it was not our intention at any time to kill any civilians. It was a commercial target, part of the ongoing war against the Brits. We offer apologies to the civilians," the caller told the Dublin office of the Irish News.

Police have said the Real IRA -- which opposes the Northern Ireland peace process and fights British rule in the province -- gave a deliberately wrong warning, driving civilians into the heart of the blast.

The Prince of Wales brings condolences to Omagh

A spokesman for Ahern said the prime minister felt the Real IRA apology was no justification for the attack, calling it a "cold comfort to the bereaved and injured."

Three known paramilitary groups, including the Real IRA, have refused to lay down their arms since feuding political parties agreed in April to share power in a Northern Ireland assembly.

The other groups are the Continuity Army Council (CAC), or the Continuity IRA, and the Irish National Liberation Army (INLA).

On Tuesday, Ahern ordered the three paramilitary groups to declare a cease-fire within 24 hours or face the wrath of his government.

David Trimble, the head of Northern Ireland's new assembly, says the bombing presents a challenge to society on both sides of the border AIFF or WAV (128 K / 10 sec. audio)

"If they don't, we will intensify every single effort that I can physically and possibly do legislatively, and with the security forces, to break these organizations," Ahern said.

Ahern said he expects to hear from all three groups before he meets in Dublin Wednesday with Protestant leader David Trimble, who serves as First Minister designate of the Northern Ireland assembly.

Ireland's Cabinet is also due to hold an emergency meeting Wednesday to discuss extra measures for clamping down on paramilitary groups.

The British and Irish governments are discussing changing laws to make it easier to convict suspected terrorists by allowing courts to accept as evidence simply the word of a police officer.

A hearse delivers the bodies to the graveyard

The grieving village of Augher, near Omagh, laid to rest a mother and her young daughter in a church graveyard on a hill -- the first in a long line of funerals for the victims.

As the funeral ended, Prince Charles arrived in Northern Ireland to bring royal condolences to Omagh.

"I'm shattered and horrified, and the least I can do is to come here and offer my sympathy and support," said Charles.

"I do have some understanding of the awful horror that people have to put up with when their relatives are killed," he added, recalling the IRA assassination in 1979 of his great- uncle, Lord Mountbatten.

During the Requiem Mass in St. Macartan's Catholic Church in Augher, the congregation, which included the victim's husband and three young children, sang, "My soul is longing for your peace."

Their 65-year-old grandmother was also killed and was to be buried Wednesday.

Traffic halted in central Belfast on Tuesday as thousands of people showed their sympathy for the victims.

The crowd gathered at Belfast City Hall to hold a 15-minute silent vigil and to listen to prayers for those maimed or killed.

They heard the city's mayor read out the names of all 28 women, children and men killed by the bombing.

At dawn Monday, police arrested five people in the attack. Under British anti-terrorist law, the five can be interrogated for up to a week before being released or charged.

The Associated Press and Reuters contributed to this report.

"The Committee: The Secret Protestant Reign of Terror in Northern Ireland:: Political Assassination in Northern Ireland" by Sean McPhilemy "The IRA: A History" by Tim Pat Coogan "Fifty Dead Men Walking: The True Story of a British Agent inside the IRA" by Martin McGartland "Ireland and the Irish: Portrait of a Changing Society" by John Ardagh "God and the Gun: The Church and Irish Terrorism" by Martin Dillon Click here for additional reading on Northern Ireland