The IRA is still an illegal organisation and must fully disband and hand over its criminal assets, the Minister for Justice, Michael McDowell said. He called for the proceeds of the December's £26.5 million Northern Bank heist in Belfast, widely blamed on the IRA, to be given back.
Speaking about the IRA's criminal assets, Mr McDowell said: "All of those things are unaffected by the cessation of the military campaign. There are matters that have to be pursued and they don't go away.
"The statement doesn't deal with issues like that but there are other issues which remain on the agenda and won't be taken off the agenda."
But Mr McDowell, - the harshest critic of Sinn Fein and the IRA in the Government, "wholeheartedly" welcomed the end of paramilitary violence and said it was a very positive step forward.
But he called on the IRA leaders to dissolve the organisation as it is illegal under the 1939 Offences Against The State Act.
"The IRA is still an illegal organisation and it must go out of existence or change so that it is no longer illegal," he said at a garda graduation ceremony in Co Tipperary.
"Its continued existence cannot be tolerated as long as it is an illegal organisation."
Mr McDowell said IRA membership was a crime and all of its assets must be forfeited to the State.
"Fundamentally its constitution is subversive of our Constitution and as long as that is the case it will remain prescribed under the 1939 Act.
"It is up to members of that organisation either to dissolve it or to change it so that it no longer infringes the condition of the 1939 Act and is no longer subversive of the Irish Constitution."
However Mr McDowell said people must be positive about today's statement.
"I think it is a major step forward that people are no longer claiming the right to use violence against the Irish Republic, that's a very positive step forward.
"I do want say that the ending of the campaign of paramilitary violence is something which I deeply and wholeheartedly welcome."
He paid tribute to the Taoiseach Bertie Ahern and the British Prime Minister Tony Blair for the "huge amount of statesmanship, work and commitment that has brought us to this point".
He said: "I feel an immense sense of pride that they have achieved in putting, insofar that it can be done, this campaign of violence to an end."
Mr McDowell said the Government has consistently stated that it's deeds not words that count. "The words were a necessary condition and we'll have to see now how it all works out," he said.