As she left for the Asian leg of her high-speed nine-capital tour, Russia continued to hold out for a legally binding charter governing its relationship with an enlarged Alliance, including a promise that no nuclear weapons will be stationed on the soil of new members.
Although Mrs Albright spoke of making "important progress" in her crucial first meeting as Secretary of State with the Russian Foreign Minister, Yevgeny Primakov, it was clear that no agreement had been reached on a number of key issues during her 24-hour trip. Although Mr Primakov complimented the talks as "fruitful", he said that Russia was "still negatively disposed" to Nato's growth, although it was "doing everything we can conceivably think of in order to minimalise the negative consequences."
However, the arrival of the Albright express in Moscow produced little of the fury that has recently emanated out of Moscow with every mention of Nato's plans to move towards Russia's borders, an issue that many believe threatens stability in Russia. Significantly, a generally amicable relationship appears to have grown up between Ms Albright and Mr Primakov - in contrast to the chill that hung over his dealings with Warren Christopher, her predecessor.
Although Nato has said it will go ahead with its expansion plans without an agreement with Russia, it would prefer to strike an accord before the Madrid summit in July, when it will start the process of admitting new members - probably, the Czech Republic, Hungary and Poland.
This will be no easy task. Although Russia can do little to stop Nato expansion, it wants to extract the best terms possible. Mr Primakov yesterday reiterated Moscow's demand that any charter with Nato should be ratified by the Alliance's member parliaments, and should include a no-nukes pledge. "Ms Albright confirmed the fact that there is no intention to deploy nuclear weapons on the territory of the countries in Central and East Europe. We feel that that should be reflected in the document," he said.
But Nato is not willing to place itself in a position where it can never deploy nuclear weapons in the region in future. And that, say Russians, means that Cold War sentiments live on.