Armed police were called late on Wednesday evening after a Norwegian man of Somali origin started to attack people in London's Russell Square, an elegant park near the site of a 2005 suicide bombing.
The victim, a U.S. citizen in her 60s, was treated at the scene but pronounced dead a short time later.
A British man, who was stabbed in the stomach, remains in hospital in serious but stable condition, police said. Four others - an American man, a man and a woman from Australia, and an Israeli woman - suffered stab wounds but have been released from hospital.
"All of the work we have done so far increasingly points to this tragic incident as having been triggered by mental health issues," said London Metropolitan Police Assistant Commissioner Mark Rowley.
"We believe this was a spontaneous attack and that the victims were selected at random," said Rowley, who is Britain's most senior anti-terrorism officer.
"So far we have found no evidence of radicalisation that would suggest that the man in our custody is in any way motivated by terrorism," said Rowley. Initially he said that terrorism was a line of inquiry.
Police, who arrived within six minutes of being called, used a Taser electric shock gun while detaining the 19-year-old suspect. He was later formally arrested on suspicion of murder.
Police cordoned off the southern part of the square, which sits at the heart of London's university area and is close to landmarks such as the British Museum, for several hours as forensics officers examined the attack scene.
Later, workmen hosed blood off the pavement.
London's Sadiq Khan, the first Muslim mayor of a major Western capital, called for vigilance and urged Londoners to report anything suspicious to the police?, who increased their presence in the capital.
"The safety of all Londoners is my number one priority and my heart goes out to the victims of the incident in Russell Square and their loved ones," he said.
Just hours before the Russell Square attack, London's police chief said that he would deploy an additional 600 armed officers across the capital to protect against attacks.
London counter-terrorism police chiefs have previously warned that Islamic State was seeking to radicalize vulnerable people with mental health issues to carry out attacks. In some operations, police commanders have taken advice from specialist psychologists.
Islamist militants hit London with coordinated suicide bombings on July 7, 2005, killing 52 people. One of the bombs detonated on a bus close to Russell Square.
Since then, dozens of plots have been foiled and there have been smaller-scale attacks, such as the beheading of an off-duty soldier by militant Islamists in a London street in May 2013.
A man who attacked passengers at a London underground train station in December was jailed for life earlier this month. The judge said the attacker was suffering from paranoid schizophrenia at the time of the offense but may have been motivated by events in Syria.