The Northern Ireland First Minister and Deputy First Minister have issued a joint appeal to the Orangemen at Drumcree "to immediately end their protest and to return to their homes".
Mr Trimble and Mr Mallon asked the Garvaghy Road residents' committee and the people of the area "to positively recognise the significance of such a move".
They held a joint press conference in Stormont yesterday afternoon to condemn the killing of the three boys in Ballymoney. A short time earlier the two men had what the Ulster Unionist leader described as "a very helpful and useful meeting" with the leaders of the Orange Order in the same building.
Mr Trimble began by saying that "three young, innocent lives have been destroyed and a family left devastated by the callous, brutal killers whose only goals are destruction and division in our society".
He then appealed directly to the Portadown Orangemen "not as First Minister but as their Member of Parliament. I know that they have tried to conduct their protest peacefully. But those responsible for these murders and for other violence have used those protests as an excuse and an occasion for an appalling act of barbarity.
"And I must say to the Portadown brethren that the only way in which they can clearly distance themselves from these murders, and show to the world that they repudiate those who have murdered young children and used them as an excuse for so doing - the only way they can repudiate that is now to leave the hill at Drumcree Parish Church and return home.
"We can come back to the issue again, come back to the question of their desire to continue and complete the traditional route at another time. But today, in the light of what has happened, the thing to do is to go home.
"As the Rev William Bingham has said - no road is worth these lives, and to go down the Garvaghy Road today or tomorrow would be a hollow event after what has happened."
The Ulster Unionist leader said he believed "we in Northern Ireland can solve our problems and that includes the problems that arise from processions. I believe we can arrive at a situation which will allow people a perfectly legitimate, reasonable expression of their cultural identities and traditions. And this can be done. And so I would say to those who have addressed you in those terms: Do not despair - we're going to get there."
Mr Trimble also welcomed the decision of other nationalist groups - such as in Keady, Co Armagh - "who in the past year or two have opposed Orange processions, and are now going to merely indicate by purely peaceful protest their opposition".
Mr Mallon said they understood that the proximity talks which began in Armagh on Saturday "will resume". Mr Trimble said that although progress in these talks on Saturday had been "agonisingly slight", he hoped that "the events of the next few days" would make progress easier.
Mr Mallon said he and Mr Trimble "between us have a mandate to strive for peace and progress for all our people. Together we go forward to find a solution to this crisis, and in the wider context, all of the other parades issues.
"There are those who have told us that's impossible. Of course they told us that it was impossible that the beginning of peace would be created. It has been. They told us it would impossible to get political agreement. It is possible - it has been done. There are those who are telling us it is not possible to get a resolution of this specific problem in Portadown and the parades problem in general. It is not impossible - we will continue to try to do it, and we will do it.
"Everyone in a position of responsibility should decide today, either we go back to the violence and the terror of the past, or we look forward to a new democracy, a new dialogue for a better future. Violence from whatever quarter has no role to play in the future of Northern Ireland."
When Mr Trimble was questioned about why he had not asked the Orangemen to disperse earlier, Mr Mallon jumped to his defence. "Let me say that I have in these past two weeks been with the First Minister for almost every waking hour. I know the attempts that have been made. I know the influence that he has used. I can see the influence he has used today. And let there be no doubt in anybody's mind that the leadership we have given in these past two weeks, we'll continue to give in the future, and is the basis upon which we're going to solve this and other problems. The leadership given by David Trimble since he became First Minister has been there for me to see."
Mr Mallon said that of three potential flashpoints yesterday, two had passed peacefully and the third, in his own constituency, was being resolved peacefully. This was understood to be a reference to parades due to take place yesterday in Pomeroy, Co Tyrone, Dunloy, Co Antrim, and Keady.
"Those things did not happen by accident. And in the rest of the coming week I hope things will not happen by accident."
The Sinn Fein president, Mr Gerry Adams, later welcomed the joint statement of the two leaders. He said that, in the aftermath of the Ballymoney tragedy, calling on the Orange Order to end the "siege" on the Garvaghy Road was "the only course of action".