The prime minister has told Gerry Adams he wants an inclusive process involving Sinn Fein - but that cannot happen as long as the IRA is active.
Martin McGuinness and Gerry Adams met Mr Blair at Chequers
Tony Blair delivered his message in "the starkest terms" on Friday.
The prime minister made the point at a meeting with Mr Adams and Martin McGuinness of Sinn Fein at Chequers.
It was the first meeting between the men since police blamed the IRA for a £26m bank robbery in Belfast in December - a charge the IRA has denied.
Mr Blair stopped short of warning that Sinn Fein would be locked out of devolution. But there are implications for the process in his words at the Chequers meeting.
The prime minister said he accepted the chief constable's view that the IRA was behind the raid.
After the meeting, a spokesman for Number Ten said: "Mr Blair and the taoiseach will be briefed further on the robbery and the investigation by the Police Service of Northern Ireland chief constable and the garda commissioner when they meet next Tuesday in Downing Street.
"The prime minister told Sinn Fein that such activity was incompatible with the Good Friday Agreement and the peace process.
Mr Blair has given a "stark" warning to Sinn Fein
"He said that while he still wanted to find a way forward which included Sinn Fein, it was his duty as prime minister to underline in the starkest terms that that would not be possible if paramilitary and criminal activity continued to be carried out by the IRA.
"A complete and verifiable end to all such activity by the IRA would be essential if progress towards such an inclusive agreement were to be possible."
Mr Adams emerged from the meeting, saying that both sides agreed that the peace process was "in profound difficulties".
He said there had been plenty of "straight talking but there had been no row."
It is understood that there are no further plans at the moment for a meeting between Mr Blair and the Sinn Fein leadership.
Earlier, Mr Adams told BBC Radio 4's Today programme he was confident the peace process could be put back on track.
"I wouldn't be here in London, I wouldn't be going to meet the British prime minister, if I didn't have both the hope and conviction that it can be sorted out. In the meantime, we are going to have to weather this particular storm."
He said he did not believe the IRA was responsible for the Northern Bank raid and he urged Mr Blair not to "get carried away with the spin of the moment".
"Tony Blair has made a huge investment in this process. He has shown leadership. This is not the time to be put off because there are difficulties. These difficulties can be overcome if there is the will to do it."
Secretary of State Paul Murphy said: "We know, as a government, that the IRA carried out this raid.
"Our view is that until that issue is resolved amongst the IRA - the whole question of criminality and giving it up - unless paramilitary activity ceases, then we simply will not be able to go on the way we have in the past."