The people of 10 former Soviet-bloc countries as well as the Turks and Cypriots are clamouring to be admitted to the most powerful economic club in the world.
By the time they leave Luxembourg tomorrow evening, EU leaders having grappled first with a dispute over Britain's exclusion from the new single currency club, are expected to have agreed an enlargement strategy which picks the cherries off the east European tree. They will invite the Poles, Czechs, Hungarians, Slovenians and Estonians, as well as the Cypriots, all of whom have been judged ready to enter negotiations on 31 March.
After years of wrangling about the danger of opening new dividing lines in Europe if they start with just a chosen few, the selective approach appears to have won out. The Swedes and Danes are set to drop demands that all three Baltic states be brought in to the bloc at the same pace.
But the deal will be packaged in such a way as to mollify the disappointed East Europeans, allowing them to claim that they are not being shut behind what the Foreign Secretary Robin Cook calls a Velvet Curtain. The Latvians, Lithuanians, Slovaks, Bulgarians and Romanians will be officially declared candidates and admitted in March alongside the frontrunners to a "screening" procedure under which their economies will be vetted against the European "acquis" - the 90,000 pages of single market legislation which all applicants have to take on board.
In theory any of them will be free to move into full negotiations as soon as they make the grade. But a longstanding dilemma over Turkey, which first applied to join the EU in 1963, could still hijack the summit. Greek Prime Minister Costas Simitis has threatened to veto plans to invite Turkey to a standing "European Conference" originally conceived as a grand launch of the enlargement process, for all applicants. It will be opened by the Queen in London in February.
Turkey's inclusion was designed to give Ankara the impression it, too, was part of the "European family" - even though few member states are comfortable about the prospect of Turkish membership because of the country's relative poverty, its size and religious make-up.
To appease Athens, the other leaders may have to agree that the standing conference will seek to tighten links between all "European" countries on such matters as human rights reform, drug trafficking and crime but make no reference whatsoever to EU enlargement. That formula would however rob the conference of any meaning since the Council of Europe already exists to promote loose links between 40 European governments.
Tony Blair, who was lobbied by the Turkish prime minister earlier this week, will be among those leaders who will insist that Turkey be given equal billing as a potential EU member. Britain considers it essential to keep alive membership hopes as a way of boosting pro-Western reformists and dampening the rise of Islamic fundamentalism.
For Ankara, a declaration from the summit of Turkey's eligibility as an official "candidate" on the same criteria as other applicants would be a victory. But the Greeks hold a veto on this and are seeking to add a further human rights condition.
Potentially even more explosive to the whole enlargement plan is its impact on the EU's hugely expensive farm and regional aid policies. Spain, the biggest beneficiary of grant aid from the bloc's structural funds, fears that the admission of countries where telephones are still a luxury will squeeze the existing poorer regions. The Spanish reject the proposal that a tight budget ceiling to cover expenditure up to 2006 be written into the Luxembourg conclusions. But Germany, despite wanting to see its eastern neighbours admitted as soon as possible, is adamant there will be no extra money beyond what is generated by a projected EU growth rate of 2.5 per cent.
The Germans, who pay most of the EU's bills, have already demanded a rebate on their own budget contributions and are resisting any radical reforms to the Common Agricultural Policy even though enlargement would double the number of farmers dependent on it.
The French, fearing that enlargement will push the EU's cumbersome decision making machinery into paralysis, want a pledge from the summit that internal reforms will be carried out before any new members are admitted. This would involve a hugely controversial shake-up to voting strengths and national representation in the European Commission.