On the constitution, although all countries now approve a "double-majority" system of member states and EU population share for most EU decision-making, the respective thresholds remained hotly contested.
Meanwhile, Germany, France, and Belgium were explicit in linking agreement on constitution to agreement on the next president of the European Commission.
Such a link complicates an already tense atmosphere.
The talks are taking place in the shadow of the failure of the first attempt to give the EU a constitution at December's summit in Brussels.
This morning, Irish Prime Minister Bertie Ahern, who chairs the summit, noted that the question of who replaces the current European Commission President Romano Prodi should be an uncomplicated matter.
"The question of the [European] Commission president is simple enough -- if somebody has a qualified majority, [has] sufficient support, then they're elected. If they don't, they won't be elected -- that's the rules, and the position is now nobody has that. No candidate has emerged yet," Ahern said.
But a candidate had in fact emerged by last night, supported by most member states -- Guy Verhofstadt, Belgium's prime minister. Verhofstadt is a liberal, with strong federalist credentials and as such strongly supported by Germany and France.
Germany's Chancellor Gerhard Schroeder made his stance plain this morning.
"I think with Guy Verhofstadt we could have a convinced European who can do the job. That is why I am sorry we have not been able to reach an accord, at least not yet," Schroeder said.
Diplomats say Schroeder last night said he would block the constitution if Verhofstadt were not chosen.
Verhofstadt's problem, however, is that too many member states oppose him. The list is topped by Britain, which does not like his federalism. The fact that Verhofstadt organized a "European defense summit" last spring together with Germany, France, and Luxembourg does not go down well either.
Officials say Italy, Poland, Slovenia, Greece, and Portugal are also against Verhofstadt.
If Verhofstadt were elected today, he is still thought unlikely to obtain the approval of the European Parliament in July.
The Irish EU Presidency has so far floated no alternative names.
Meanwhile, constitution talks have hit their own snags. Most of them have to do with disagreements over "double-majority" thresholds -- the question of the number of countries and share of EU population needed to approve, or block, a majority decision.
Ireland today tabled a complicated compromise proposal, suggesting that the member states threshold be set at 55 percent and the population requirement at 65 percent. However, on a range of questions to do with foreign policy, justice and home affairs, economic and monetary policy, etc, the member state requirement would be set at 72 percent, while the population threshold would remain at 65 percent. Critics have already pointed out this would mean making decisions would be harder than under rules set out in 2000 in the Treaty of Nice.
Ireland's foreign minister, Brian Cowan, however, argued that the most important benefit of the new system is that it introduces the "double-majority" principle. This, he says, is a clearer way of reaching decisions, compared to the largely arbitrary way votes were divided among member states under the Treaty of Nice.
Up to 13 smaller member states continue opposing the Irish proposal, asking for the member state ceiling to be set at 58 percent. Their main concern is the perceived edge the proposal would give to the largest member states.
Nerves began fraying this afternoon as differences persisted and some countries threatened to open other questions, thought to be already settled. France, for example, was said to want to reexamine whether certain tax issue could be exempt from national veto rights.
A number of countries led by Britain insist that vetoes be retained in all of tax and social policy, as well as foreign and defense policy.
Officials say it is impossible to predict how long the summit will take to finish. They say a possibility exists that the EU will have to call an emergency summit in July to elect the commission president if no deal is reached today.
Luxembourg's Prime Minister Jean-Claude Juncker is thought to have best chances then. He currently refuses to run for the office.