Mr Bruton appealed to the bombers to end their campaign well before the deadline for the all-party talks. He said the attempt to blow up the Hammersmith Bridge in London was "beyond me".
He told the Independent in an interview in Dublin that the IRA now had the chance with Sinn Fein to seize what they had been demanding - entry to all-party talks in Northern Ireland. He did not believe the logic of their position would allow them to throw it away by continuing the London bombing campaign. "The logic from the point of view of the republican movement is they now see an opportunity they have been after for years is now available to them."
Behind Mr Bruton's optimism, hard bargaining is going on between London and Dublin over how to move the talks forward if the IRA restore the ceasefire, allowing Sinn Fein to participate in the negotiations.
Senior officials said the British feared the IRA could leave the announcement until the day before the all-party talks were due to begin and they have questioned whether Sinn Fein could be admitted if there was lack of confidence that the second ceasefire would hold.
Irish officials who met British counterparts yesterday insisted that Sinn Fein must be admitted if there is to be a ceasefire. Mr Bruton said: "These talks start with the whole agenda. When you look at the logic of that, I think there is grounds for assuming intelligent and thoughtful people will see that the opportunity they have been seeking is now accessible to them in a way that has never been accessible to them before.
"The sooner the ceasefire is in place before the talks the better from the point of view of a better atmosphere.
"How you can explain the Hammersmith bomb in that context is beyond me. It is political erroneous and morally unjustifiable."
The Irish Foreign Minister, Dick Spring, meanwhile urged the British government to hive off the issue of decommissioning IRA weapons from the main political talks on 10 June.
Mr Spring told a meeting of the inter-parliamentary body of British and Irish MPs in Limerick: "There is a strong argument in favour of remitting this issue in the first instance."