The move will breathe new life into a peace process which many had considered moribund since the last cessation broke down in February 1996 with the IRA bomb which devastated Docklands in London.
Ministers were surprised by the speed of events. Having held an Anglo- Irish summit with Ray Burke, the Irish Foreign Minister, at Westminster, Mo Mowlam, the Secretary of State for Northern Ireland, had gone to her constituency in Redcar.
Mr Blair was in Wales for the public relations offensive for the Welsh referendum on devolution. The Prime Minister and Ms Mowlam spoke last night on the telephone. "We are all just waiting to hear the words from the IRA," one Whitehall source said last night.
In Washington, the Clinton administration welcomed the Sinn Fein initiative. A White House statement said last night: "We warmly welcome the news that the Sinn Fein leadership has urged the IRA to call a ceasefire and we very much hope and urge the IRA to implement a lasting ceasefire."
But while the expected ceasefire will be seen by many as an historic new opportunity for progress, it will also pose big difficulties. The most immediate of these is the question of whether the main Unionist party, David Trimble's Ulster Unionists, will remain at the talks table if Sinn Fein is granted entry in mid-September.
Mr Trimble is to meet Tony Blair on Monday for the second time in a week to try to persuade the Government to toughen its position on the controversial and long-running issue of arms decommissioning. There seems no likelihood, however, of the Government shifting its ground.
The decisive factor in the cessation move appears to have been Mr Blair's decision to abandon John Major's previous stress on decommissioning. Republicans have long insisted that a new IRA ceasefire could only come about if Sinn Fein was guaranteed a place at the talks without decommissioning.
The Ulster Unionists, dismayed by Mr Blair's stance, have subsequently adjusted their own position. But the party believes itself vulnerable to attack from other Unionists, in particular the Rev Ian Paisley, if it consents to sit down with Sinn Fein if other Unionists absent themselves.
The first firm public indication of a ceasefire move yesterday came when Sinn Fein leaders Gerry Adams and Martin McGuinness announced that they had urged the IRA to reinstate the 1994 cessation. They said they had given the IRA a detailed report and assessment of the situation, and the IRA had assured them it would respond without delay.
Mr Adams added: "I have made it clear over the 18 months since the collapse of the peace process that I would only approach the IRA to restore their cessation if I was confident that their response would be positive." This was seen as the broadest of hints that a ceasefire was on the way.
A hint had come in a joint statement from Mr Adams and John Hume, the SDLP leader, the first for some time, which struck an upbeat note. The two leaders said progress had been made, declaring themselves "optimistic that outstanding obstacles to inclusive negotiations in a peaceful atmosphere could be removed".
The republicans may have decided that a new political landscape has been created with the election of Mr Blair and that in Dublin of a new Fianna Fail Taoiseach, Bertie Ahern. Both leaders are viewed as more open and receptive to the peace process than their predecessors, John Major and John Bruton.
Mr Ahern warmly welcomed the moves. He said: "I hope and trust that the response of the IRA will be positive, and that we'll be able to get back into ceasefire and able to plan to move to all-party talks ... I just hope that we can give Mr Trimble the necessary guarantees."
However, an unwelcoming note was struck by Jeffrey Donaldson, the Ulster Unionist MP who is also a leading Orangeman. He said he was not entirely surprised by the move since Mr Blair had made a point of giving republicans "virtually everything they ever asked for".
The deputy leader of the Rev Ian Paisley's Democratic Unionists, Peter Robinson, asked rhetorically: "Why would the IRA not declare a ceasefire? They have got everything they have asked for."
Clouds of mistrust, page 6
The rocky road
9 February 1996: IRA bombs London Docklands, ending 17-month ceasefire, killing two and injuring more than 100.
15 February 1996: IRA member Edward O' Brien blows himself up as bomb detonates prematurely.
30/31 May 1996: Peace Forum elections in Northern Ireland. Sinn Fein gets 15.74 per cent of vote.
15 June 1996: IRA bombs heart of Manchester injuring about 200 people.
July 1996: Drumcree stand-off causes widespread rioting in province after initial blocking of march is lifted. Taxi driver Michael McGoldrick is killed by loyalist paramilitaries on 8 July.
7 October 1996: IRA plants two bombs at Thiepval barracks, Lisburn. Bombs injure 31. Warrant Officer Jim Bradwell dies five days later from injuries.
21 December 1996: Loyalists blamed for booby-trap bomb under car of leading republican Eddie Copeland.
12 February 1997: Lance-Bombardier Stephen Restorick shot dead by an IRA sniper at a checkpoint in Co Armagh.
March/April: IRA begins bomb campaign in run-up to general election. Grand National postponed after bomb warnings.
May: Labour elected Sinn Fein increases vote. Wins two seats. High-level talks held between Government and Sinn Fein.
16 June: Tony Blair bans all further contact between senior civil servants and Sinn Fein representatives after the IRA guns down RUC officers John Graham, 34, and David Johnston, 30, in Lurgan, Co Armagh.