It's 1961. With tension between the superpowers at its height, the Soviets are attempting to rush their untried new atomic submarine, the eponymous K-19, into service.
When the craft's commander, Polenin (Neeson), objects to the perilous haste, politically well-connected martinet Captain Vostrikov (Ford) is appointed over his head.
With Polenin on board as second in command, K-19 begins its maiden voyage.
Determined to deliver or drown, Vostrikov pushes both creaking sub and mutinous crew to their limits.
K-19 successfully launches its test missile and heads for a patrol station 400 kilometres from the US coast.
But the sub's nuclear reactor cooling system has developed a leak. As the reactor core heats up relentlessly, the hopelessly ill-equipped crew put their lives on the line to avert disaster, conscious that a nuclear explosion so near the US could tip the superpowers into mutually assured destruction...
Just how far we have come from the dark days of the Cold War is shown by the fact that this tale of Soviet heroism is being told by an American studio.
The film's principal antagonists are, of course, played by Hollywood stars, but they both attempt Slavic accents - Neeson with rather more success than Ford, it has to be said.
But the movie's true hero - or heroine - is Bigelow, who once again proves herself one of the very best action directors around.
In such earlier films as "Point Break" and "Strange Days", she dazzled viewers with her darting, breathless travelling shots.
You'd think being confined to a submarine would cramp her style. Far from it.
She fluently manoeuvres her camera from one end of the sub to the other, evoking a clammy claustrophobia while cranking up the tension to an almost unbearable degree.
"K-19: The Widowmaker" opens in UK cinemas on Friday 25th October 2002.