The Taoiseach said the Government would give full support to the Garda in cracking down on the "Real IRA".
Mr Ahern said that it would take the security forces in Britain, the North and the Republic to thwart their efforts.
"We will continue to do our utmost to give the room and resources to the Garda to do that."
Mr Ahern said he would like to think they had seen the end of all kinds of terrorism, but he believed, from all the briefings he had received, not just at the weekend, that there were still certainly some units of the "Real IRA" in Britain.
"There is some reason to believe that it is the same unit or two for some considerable time. They have not been thwarted as yet.
"They are not going to listen either, unfortunately, to all our efforts to stop them in their tracks. Hopefully, they will be apprehended as soon as possible. We have had more success here to date.
"There is a dangerous, violent group which is determined to continue on, to make life difficult for everybody.
"I think some of the comments by some of the British media, over the past few days, that perhaps Sinn FΘin could control them, are totally wrong," Mr Ahern said.
The Taoiseach was replying to the Fine Gael leader, Mr Michael Noonan, who said the dividend for everybody on the island was peace.
"If peace is not fully delivered, then people's political positions will be undermined once more. And in that context, will the Taoiseach tell us if the bomb in Birmingham, which was attributed to the "Real IRA", was a once-off event or has he been briefed on the basis that there may be an ongoing campaign of violence, both on this island and in the United Kingdom?"
There was a cross-party welcome for the election of Mr David Trimble as First Minister and Mr Mark Durkan as Deputy First Minister.
Mr Ahern said he hoped there would be a speedy resumption of the work of the Executive, the North-South bodies, the British-Irish Council and the other institutions of the Belfast Agreement.
"Hopefully, they can now move to a position where they can work to May 2003, which is the mandate of the institutions to achieve a fair amount of political progress."
