The most important issue she faces is tension between Nato and Russia. This is a greater immediate threat to global security even than the long- term issues of water shortage, population growth and global warming. Whatever its troubles, Russia is still a nuclear-armed power which is heir to 85 per cent of the military strength of the Soviet Union and still dominates the Eurasian landmass. Nato is planning to enlarge to the east. This upsets the Russians.
The United States, with Britain in tow, wants Nato to embrace the new democracies of central Europe. Mrs Albright is the personification of the forces operating on American politics. One of the candidate members is her homeland, the Czech republic. The other likely candidates, Hungary and Poland, also have powerful advocates among the immigrant communities of America. So Mrs Albright wants the next Nato summit in July to issue formal invitations to these three countries to join the 16-nation North Atlantic alliance. If that happens, they are expected to become members on or before 4 April 1999, Nato's 50th anniversary.
Russia does not like it. Jacques Chirac, who talks to Mrs Albright today, urges caution. So is Nato enlargement a good idea? Before we answer that question, we have to ask a more basic one: What is Nato for?
Nato is a military alliance and a highly successful one - it won the cold war. When the Soviet Union broke up, so did the Warsaw Pact alliance, Russia's cordon sanitaire. There would have been a certain logic in Nato disappearing as well. But instead, former members of the Warsaw Pact wanted to join Nato and Nato agreed that they should.
The motives on both sides were obvious. The central Europeans saw Nato membership as another credit card that free capitalist democracies carried in their wallets. They really want to join the European Union, but that is more complicated and will take longer. Admission to Nato is seen as a first step. Nato itself, meanwhile, was an institution in search of a role. And there is some evidence that, since the end of the cold war, it could serve a purpose. It has a formal role in Bosnia, and even though the Gulf war was outside its area, Nato membership meant America's allies talked the same language and used compatible equipment and procedures.
But, equally, the Russians have good reason to regard Nato expansion as a threat. The war of words became heated last week when Russian officials reiterated Moscow's 1993 Military Doctrine, that it might use nuclear weapons in response to a conventional attack.
So, is the rush towards Nato's expansion premature? The first thing to be said is that there are real obstacles in the way, even if the US thinks it can use its diplomatic muscle to push them aside. Nato enlargement has to be ratified by the legislature of all 16 Nato members, including two-thirds of the US Senate. Turkey has already threatened to refuse to ratify new members' accession if its ambitions to join the European Union are frustrated, as they will continue to be.
There are important questions about whether the Czech, Polish and Hungarian armed forces are ready to be integrated into Nato systems.
But Nato enlargement is not just about matters military. The candidate members know that. And the US sees it that way, too. A senior US diplomat earlier this month said he saw Nato enlargement as a "second bite at the apple", in trying to complete the 1947 Marshall Plan, extending post-war reconstruction to the countries frozen out by Stalin.
However, the Prime Minister should warn Mrs Albright that this laudable aim should not be pursued to the extent of freezing Russia out. If Nato does have a role in the post-cold-war world, it should be to help co-ordinate responses to all threats to the rule of international law. If that is the aim, then present plans for expansion are potentially dangerous because they are too modest, in that they do not include Russia and other nuclear powers.
The argument against more ambitious expansion is that it would weaken it by making Nato too diffuse. But Nato was always diffuse. Its guarantee spanned the Atlantic, linking North America and Europe and the promise that an attack on one would be regarded as an attack on all applied to all members, including Belgium and Iceland. It always had members who were militarily weaker than others - notably Greece and Turkey. But their membership was important for political and strategic reasons. The principle of common security bound the members together. On that basis, Nato enlargement should go ahead. But - and the but is becoming bigger - it must be accompanied by a firm agreement which pulls Russia in from the cold and binds it firmly into the European security system.