These undecided Unionists look set to have a decisive effect on the result of the poll. While almost all other voters in Ireland, north and south, have decided how to vote, more than 40 per cent of Mr Trimble's party is either unwilling or unable to give their views to opinion pollsters.
Their final decision is likely to determine whether the "Yes" lobby receives a convincing endorsement, or whether Unionism will be seen to be hopelessly split on Northern Ireland's future.
An opinion poll in the Irish Independent found 44 per cent of UUP supporters favouring the agreement, 14 per cent against and 41 per cent undecided.
Those in this undecided category were targeted yesterday by both Tony Blair, the Prime Minister, and William Hague, the leader of the Conservative Party, who flew to Northern Ireland to make a last-ditch attempt to increase the "Yes" vote. Mr Blair argued in a speech last night that Unionists had nothing to fear and everything to gain from the Good Friday agreement.
He has also issued a handwritten poster setting out a number of pledges which promise fairness and equality for all, that those who used or threatened violence would be excluded from government, and that prisoners would be kept in unless violence was given up for good.
Mr Blair appeared with Mr Trimble, who is expected to become first minister in a new assembly following the referendum. The Prime Minister was speaking at the University of Ulster in Coleraine, Co Londonderry.
Mr Blair said: "People feel an enormous sense of responsibility. I don't know whether we'll ever have a chance like this again in a generation."
Turning to Mr Trimble, he continued: "He has shown courage, tenacity and leadership that has been missing from the politics of Northern Ireland for a long time and has been found now in David Trimble."
Mr Trimble yesterday claimed that the tide was turning in favour of the "Yes" campaign.
The Secretary of State for Northern Ireland, Mo Mowlam, reinforced this message during a walkabout in Belfast with the Virgin chief, Richard Branson, who said he wanted to bring more business into Northern Ireland and predicted that peace would bring "an enormous amount of inward investment".
He added: "If it goes the other way it will set everything back and things will perhaps get worse."
One of the main apprehensions about the peace agreement among Unionists have been about the release of prisoners convicted of terrorist offences. Yesterday, a group of former loyalist prisoners, Billy Mitchell, Tom Winston and James Tate, appeared at a "Yes" campaign press conference to try to allay fears of fellow Protestants.