At a personal level too, the omens are set fair. He seems to get on famously with his opposite number Madeleine Albright, who shares his moralistic, and moralising, approach to foreign policy. But the trip offers Mr Cook more than just a chance to strut the corridors of global power. For reasons having little to do with Britain, this Presidency could mark a new departure in the relationship between the EU and the US.
Up to now, that relationship - institutionally at least - has been a shadow of what it should be. Barring a trade war, summit meetings between the presidencies of the EU and the United States have been international diplomacy's great invisibles. When held in the US, they were deemed by the White House of about equal importance to a protocol drop-by at the Oval Office by the President of Vanuatu - and by the US media as about as newsworthy. In Europe, they tend to be cobbled into a few spare moments on the sidelines of a G-7 summit. And the barely disguised lack of interest by Messrs Reagan, Bush or Clinton was entirely understandable. Trade apart, they knew real power on the old continent still lay not in Brussels, but in Paris, Bonn and London. The alliance that mattered was military. When push came to shove the European who counted in Washington was the one who was Secretary-General of Nato.
But, I suspect, not for much longer. Perspectives are starting to shift. For one thing economics and technology, strong suits of the EU, increasingly drive world affairs - and the financial shambles in Asia has been a blast of cold reality for those arguing that America's future straddled the Pacific, not the Atlantic. Just a couple of days ago, Richard Holbrooke, America's in-house diplomatic bulldozer, warned of a "very rough year" ahead for American foreign policy. He wasn't talking about ringgits, bahts, President Suharto or even China - but security flashpoints in Europe and its periphery; from Yugoslavia, through Greece, Turkey and Cyprus to the Middle East and Iraq, where confrontation with Saddam Hussein may well lead to a "significant use of force" - Holbrooke-speak for something rather more than a few Cruise missiles fired in the direction of Baghdad. Europe may be self-obsessed, stodgy and underperforming. But right now it looks a wiser and safer bet than mendicant Asia.
Second, the advent of the Euro, whose launch arrangements will be largely settled during the British Presidency, will transform the EU. If the single currency succeeds, the dollar will have a serious rival as a global financial vehicle. There will be a European central bank, independent of national policymakers and with at least as much clout as the Federal Reserve. Other common institutions are bound to follow. EU enlargement will lock former Soviet satellites more closely into Europe than Nato membership ever will. Simultaneously, if a Union of 20 or more is to function, individual capitals will have to surrender powers of veto, further curtailing their ability to defend purely national interests.
So much for the big picture. On the details too, concerns are moving closer. Washington and Teheran are gingerly seeking to end their private cold war: not as quickly as as Europe would like, but quickly enough perhaps to blunt the US sanctions against Iran and all who do business with her, which so infuriate the Europeans. Both have a massive interest in keeping the lid on tensions between Greece and Turkey, and in its handling of entry negotiations with Cyprus, the EU will - whether it likes it or not - have a vital part to play. And finally the Middle East, where a stronger European involvement might just allow what is left of the "peace process" to be revived.
The fly in the ointment to all this? Paradoxically, it could be another diplomatic formulation that is already undergoing a revival. I refer to the "special relationship". The term was much bandied about when Mr Clinton paid his impromptu visit to London last May to bask in the reflected glory of the newly elected Tony Blair and underline the ideological and personal affinities of New Democrats and New Labour. We will be hearing even more of it when Mr Blair goes to Washington in three weeks time, for what promises to be a touchy-feely extravaganza of Anglo-American harmony.
All of which however risks rekindling old British illusions - and eternal European suspicions that Britain's commitment to Europe is less than the current holders of the Presidency would have us believe. The doubts exploded last month with the row over Britain's demand for a place in the inner councils of the single currency it is not joining. So in these heady times, let Messrs Cook and Blair not forget what the Americans (and they themselves) love to remind us - that Britain's importance to the US is a function of its influence within Europe. In Washington too, the EU's hour has come.