Mr Trimble, the Ulster Unionist leader and First Minister, said the meeting had been "civilised and workmanlike", while the Sinn Fein president described it as "constructive and useful, a good meeting".
Meanwhile, in a significant relaxation of security, the Royal Ulster Constabulary said all army patrols were to be taken off the streets of Belfast at the weekend.
Although there were no handshakes at the meeting, which took place at Stormont within the context of the new assembly, it was described as cordial. While many political battles lie ahead, some optimistic observers were predicting it could help set a new tone.
The meeting had two phases. In the first, members from each party discussed various technical and administrative matters concerning the assembly. In the second phase, Mr Trimble and Mr Adams went into another room together for more wide-ranging discussions.
These touched on IRA arms decommissioning, a matter raised by Mr Adams, but even this issue, which has generated so much heat in the past, appears to have been discussed without any raising of voices.
Afterwards, both men conspicuously refrained from launching attacks on each other, although Mr Trimble did observe wryly that in politics "you have to meet and work with a range of people and that they are not all angels".
But even in dealing with decommissioning his tone was markedly conciliatory. He said: "We are not saying to people that you have to surrender arms or in any way feel humiliated. I am quite satisfied in the discussions I have had this morning that Mr Adams recognises the need for progress to happen on all fronts. I am sure he is aware of the obligations of the republican movement on the front of decommissioning. I'm encouraged and I hope that things will happen."
Mr Adams said of the Unionist leader: "He's a man who I can do business with; he's a man I have do business with. I put it to Mr Trimble that he and I had been cast in these positions and we had to narrow the gap. We had to find ways for him to help me and me to help him." In a later television interview Mr Trimble responded with a grin: "I hope very much that Gerry will do the business."
The next milestone in the peace process is expected today, when about half a dozen inmates are scheduled to be freed from the Maze prison. The men will be the first to be freed under the early-release scheme laid down in the Good Friday Agreement.
Announcing the removal of troops from Belfast, Ronnie Flanagan, the RUC Chief Constable, said he wanted to move as quickly as possible to the position where there was no need for any military patrols anywhere.
Echoes of the past, page 4
Review, page 3Reuse content