The Oscar-winning actress will take the role of Broadway star Margo Channing, originally played by Bette Davis.
The production will open in Spring 2018, with further details to be announced.
Ivo van Hove, whose recent plays include Hedda Gabler at the National and Obsession at the Barbican, will direct.
The last time she was on stage in London, Blanchett was hailed as "mesmerising", "magnificent" and "beyond terrific".
That was in 2012, in a play called Big And Small (Gross und Klein) produced by the Sydney Theatre Company.
Earlier this year, Blanchett received rave reviews for her Broadway debut in New York in The Present, a reworking of Chekhov's play Platonov.
The plans for All About Eve were confirmed by Sonia Friedman Productions and Fox Stage Productions.
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Cate Blanchett is to return to the London stage in a new adaptation of the classic 1950 film All About Eve.