That though will be the easy bit. The nitty-gritty negotiations to marry the rich economies of Western Europe with their formerly Communist and much poorer Eastern sisters will be tough enough. At least as hard, however, will prove the task of reforming the Union as it currently exists. For the simple truth, which to his further credit Mr Cook also pointed out, is that before Europe takes new members aboard, it must first put its own house in order.
Basically, the European Union is still run according to the rules laid down in the Treaty of Rome which set up the original European Economic Community in 1957. Since then the Six have become Nine, 10, 12 and now 15. Along the way, structures and policies have been periodically tinkered with. For 40 years, the engine has just about kept on the rails, albeit grinding ahead more slowly with each new carriage that has been hitched to it. But a Europe of 21, not to mention one of 26, will surely drive our wheezing 40-year-old model into overdue retirement. For an expanded Europe to function at all, changes will be required across the board - from the Common Agricultural Policy and the allocation of regional subsidies, and above all in the EU's institutions. And these changes, at least as much as the other great project of the hour, the single currency, will reveal just how supranational the future Europe will be.
Take the Brussels Commission, the EU's ideas factory, executive and mediator rolled into one. With 20 members, it is already too big now. Enlargement, implying at least six extra Commissioners, would render it even more unwieldy and incapable of decision-making. If it is to be streamlined to say 10, countries would have to drop their claim to at least one Commissioner apiece. The best solution would be a Commission president approved by all member governments, who then picks his own team - either without regard to nationality, or on a loose regional basis.
Ditto the Council of Ministers, the ultimate seat of power in the EU, whose chair Britain assumes in January. Already the Presidency is a massive organisational burden for smaller countries, and the case for groups of countries sharing the task becomes steadily stronger. And Mr Cook's grandiloquent undertakings this week do not mask another problem. Britain will be running the show for only six months, and will not have another turn until 2004 or 2005. In the meantime other countries, some of them undoubtedly less keen on enlargement, will be in charge. What price then a long-forgotten "flying start" at Buckingham Palace?
Then there is the matter of the extension of majority voting, something European leaders singularly failed to agree at their Amsterdam summit in June. The current right of veto to protect "vital national interests" already means progress is arduous enough. An unreformed Europe of 21 or 26 - even one day 30 or more - would resemble the UN without the Security Council. Something has to give, if the institution is to continue to function. The choice lies between some form of weighted majority voting that is binding or, more probably, a generalised right of "opt out" - in other words a multi-speed Europe in which the convoy need not move at the speed of the slowest ship, and where on particularly voyages, some ships need not leave harbour at all.
This is precisely what is happening with the single currency. However debatable the euro's economic merits, it is a vastly simpler project than enlargement. Why? For one thing, it will have at most 11 members to begin with, all committed to an agreed goal. Second, because they are united on the objective, they have found it easier to surrender sovereignty - not just to an independent central bank, but to one on whose board they are not even guaranteed a seat. The horsetrading beforehand (who will be president of the bank, its links with political authority) is fierce. But when the deal is done, there will be no turning back.
All of which leads to the final and most important consequence. The changes mooted above are both essential, if an expanded Europe is to work properly, and momentous. Properly executed, they should be a step towards the "people's Europe" beloved of Messrs Cook and Blair, a Europe more accessible and comprehensible to the average citizen, doing things that have a tangible impact on his daily life. But perforce, this Europe will be more supranational. It must therefore be more accountable and democratic. The obvious, but controversial means of achieving this is to increase the powers of the European Parliament, the EU's one directly elected body, whose members are currently more celebrated for their alleged wizardry with expense accounts than for their legislative bite. But if not the European Parliament, then what ?
Such are the preconditions of successful EU expansion. Ultimately of course the option remains of doing nothing, of giving the negotiations a rousing ceremonial send-off in London, and then quietly ensuring they go nowhere. But as Mr Cook's talking this week underlines, that would be worse than funking the issue. It would be a betrayal.