David Trimble leads a party whose voters, according to this poll, strongly support the Belfast Agreement and want the all-party executive formed by July 18th.
Mr Adams's party's voters also overwhelmingly support the Belfast Agreement and say they want decommissioning to go ahead as planned, the poll results indicate.
Of course, Mr Trimble's party colleagues may not agree with their voters, and the IRA's army council has yet to make it clear that it agrees with Sinn Fein's voters.
But for supporters of the British-Irish plan to break the impasse over the executive and decommissioning, these Irish Times/MRBI opinion poll results are very positive.
First, Mr David Trimble will take encouragement from the outcome. Despite very vocal unionist opposition to the proposals put forward at Stormont by the British and Irish governments last week, a small majority of unionists supports the formation of an all-party executive on July 18th, as proposed in that document.
More crucially, there is a 2-1 majority among UUP supporters in favour of the executive being formed, according to the poll. They support it despite having received no overt encouragement from Mr Trimble to do so, and having received loud discouragement from many prominent UUP figures.
If they support the deal so clearly without any leadership encouragement, they would undoubtedly do so by an even greater margin if Mr Trimble attempted to lead them to do so. Mr Trimble's party may not be open to being led to support what is on offer: his party's supporters clearly are.
Republican voters also show substantial support for what to the republican movement is the unpalatable part of the deal, decommissioning.
Some 65 per cent of Sinn Fein voters believe decommissioning should go ahead as planned, with just 26 per cent saying they are opposed.
In addition, 60 per cent of Sinn Fein voters believe the IRA should make a statement indicating its support for the proposals made by the British and Irish governments, with 28 per cent opposing this idea.
The unionist support for the executive being formed comes despite marked pessimism among unionists about the prospects for decommissioning. Just one in 10 unionists believe decommissioning will be completed as planned by May 2000, with 71 per cent believing it will not.
Just 23 per cent of unionists believe Gen John de Chastelain's independent international commission can eventually bring about total decommissioning with 56 per cent believing it cannot. The British and Irish governments will take encouragement from the fact that UUP and Sinn Fein voters appear willing to accept the parts of the package which the leaderships of the two parties find difficult.
If the British government is seeking a means of bolstering unionist support for the plan, the poll shows that the so-called "failsafe" mechanism is the key. A total of 53 per cent of unionists believe the proposal that the institutions already established will be suspended if decommissioning does not take place is an insufficient safeguard, with just 32 per cent believing it is.
Indeed, unionists are not on their own in objecting to the notion that the executive would be suspended in its entirety if one party does not abide by the terms of the agreement. The proposal that the executive should continue without parties which do not abide by the terms of the agreement is endorsed by a majority of more than two to one.
The nationalist population is evenly divided on the question, although close to a 2-1 majority of SDLP voters also supports the continuation of the executive if one party defaults on its commitments.
Unsurprisingly, 81 per cent of unionists support the continuation of the executive without an offending party, with just 11 per cent against.
Oddly, there is more approval among those polled in the North for the handling of the agreement by the Irish Government than by the British government.
Overall, 65 per cent of those polled in the North support the formation of the all-party executive, and 75 per cent said they believed decommissioning should go ahead as planned.
There is, as expected, even greater support for the plan in the Republic, with 84 per cent believing the executive should be formed and 85 per cent saying decommissioning should go ahead as planned. Some 82 per cent in the Republic believe the IRA should make a statement indicating support for the plan of the two governments.
Those polled in the Republic were fairly evenly divided when asked about their level of confidence that the Belfast Agreement would actually be implemented. Just 9 per cent said they were very confident, 42 per cent were fairly confident, 35 per cent not particularly confident and 10 per cent not at all confident.
The Government has come well out of the recent intense negotiations, with 86 per cent in the Republic saying they are happy with the way the agreement has been handled by the Government and just 9 per cent unhappy.
The British government's handling of the agreement receives almost as much approval in the Republic with 81 per cent approving of it and just 12 per cent disapproving.
Finally there is an indication of a modest slippage in support for the Belfast Agreement in the North, although there is still close to a 3-1 majority in favour.
However, a significant number of unionist supporters polled say they have turned against the agreement, with 52.4 per cent indicating they voted for the agreement last year and 39.3 per cent saying they voted against.
Now 38.7 per cent of unionists say they would vote for the agreement if there was another referendum with 47.8 per cent saying they would vote against.
Among Mr Trimble's party supporters, support for the agreement remains solid with 55 per cent saying they would vote for the agreement again, and 30 per cent saying they would vote against.