The proposal is the firmest expression yet of a determination to make Russia feel it has a special relationship with the West and allay fears about Nato enlargement. But it also shows that the strident attacks on the alliance's enlargement in Moscow are finding their mark.
The Secretary of State emphasised Nato's determination to take in new members from Eastern and Central Europe as as soon as possible. She asserted: "Russia will have a voice but Russia will not have a veto."
Since the end of the Cold War, a special relationship between Nato and Russia, known as "16 plus one", has evolved. But Nato has refused to entertain the idea of a Russian veto over Nato decision-making .
Ms Albright suggested troops from one or more of Nato's 16 nations, and from Russia, should live, work and possibly fight together in a single, multinational brigade. It is a radical suggestion, intended to dispel suspicions in Moscow that Nato is a hostile organisation. "I would hope that it would eventually have a genuine purpose ... and not be a showcase kind of a thing," she said.
Ms Albright's suggestion and a proposal for a Nato-Russian accord are part of a strategy to woo Russia before the Madrid summit in July, but neither looks particularly convincing. Although the US is determined to push Nato expansion forward, the obstacles remain real.
Russian officials have reacted strongly to Nato's plans to expand to embrace East European countries and last week mounted a war of words against Nato's Secretary-General, Javier Solana, as he toured the former Soviet republics of Moldova, Georgia, Armenia and Azerbaijan.
But opposition is also building from other quarters, with press comment in the US and Europe increasingly hostile to the idea. It is clear Ms Albright faces an uphill struggle.
In her first speech to Nato since her appointment as Secretary of State, she said Nato's vision of a joint Nato-Russia council "would promote a regular dialogue on majority security issues, reach concerted decisions whenever possible, and seize opportunities for joint action". Mr Solana said he hoped Russia and the alliance could seal a new relationship at a special summit ahead of the Madrid summit.
Nato also agrees to Russian demands for changes in the negotiating terms of the conventional arms treaty (CFE), to reassure the Russia that Nato has no intention of stationing forces on the territory of its new member states in the East.
Diplomats last night responded positively to Ms Albright's suggestion. A Russian airborne brigade of 1,500 troops has been deployed in northern Bosnia for more than a year, and has been working under the control of the US headquarters in Tuzla. In November last year Russian and US troops worked together to prevent a large scale incursion into Serb territory by armed Muslims.
Nato sources said a joint Nato-Russian force would not be out of the question. There are, however, problems caused by the different ways Russian and Nato units are organised.
Ms Albright arrived in London last night. But the toughest part of the European leg of her trip comes tomorrow, when she flies to Moscow.
Albright's script, page 10
Letters, page 13
Portrait, page 14