At a hearing in London's Old Bailey court, Zakaria Bulhan, 19, admitted killing 64-year-old retired teacher Darlene Horton.
He also admitted charges of wounding with intent with respect to Martin Hoenisch, Lillie Selletin, David Imber, Bernard Hepplewhite and Yovel Lewronski.
The August 3 stabbing spree initially sparked fears it was a terror attack as it took place in Russell Square, the site of one of the suicide bombings in 2005.
Police quickly arrived at the scene and arrested him.
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Bulhan, who lives in south London, is a Norwegian national of Somali origin who came to Britain as a young child in 2002.
At the hearing, the court heard how the suspect was suffering from an "acute" episode of paranoid schizophrenia at the time, with prosecutor Mark Heywood saying he accepted Bulhan's plea.
Describing the attack, Heywood said the suspect was seen "moving in an erratic fashion".
"Without warning or provocation, the man stabbed six people in relatively quick succession, saying nothing to any of them, moving on after each stabbing towards his next victim," he told the court.
Horton, a mother of two, was stabbed in the back, with the blade piercing her left lung and her heart. She died hours before she was due to fly home.
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She was married to a professor at Florida State University who had been teaching summer classes in London.
The three men injured in the attack were a Briton, an American and an Australian. Two women, an Australian and Israeli, were also wounded.
Located in central London, Russell Square is close to both the British Museum and the University of London and has a number of popular tourist hotels.
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