Scottish nurse Pauline Cafferkey, who contracted Ebola in Sierra Leone, is showing signs of improvement and is no longer critically ill, the Royal Free Hospital in London said.
The Scottish public health nurse remains in isolation at the hospital where she is receiving specialist care.
She was diagnosed with Ebola after returning from Sierra Leone to Glasgow and was initially admitted to the city's Gartnavel Hospital on December 29, then transferred to the Royal Free Hospital the following day.
The hospital said: "The Royal Free Hospital is pleased to announce that Pauline Cafferkey is showing signs of improvement and is no longer critically ill. She remains in isolation as she receives specialist care for the Ebola virus."
Prof Hugh Pennington, one of Britain's leading microbiologists, described the update on Ms Cafferkey's condition as "excellent news".
Although there are no specific details on her condition, he said it was "reasonable to assume that she was very close to being in a state of shock".
"This can mean problems with blood pressure, difficulty breathing and organ failure," Prof Pennington said.
"All these sorts of things can be dealt with in an intensive care unit with artificial supports to keep someone going.
"When you're into that kind of situation, every minute of every day is critical."
He added: "Although we don't know the specifics of her case, it is reasonable that the interpretation of what they are saying is that she is no longer in need of any of these artificial supports.
"It is excellent news and suggests she has turned a corner."
Ms Cafferkey, from Cambuslang in South Lanarkshire, had volunteered with Save the Children at the Ebola Treatment Centre in Kerry Town, Sierra Leone, before returning to the UK.
Save the Children has launched an investigation into how she was infected but admits it may never establish the exact circumstances.
She flew back to the UK via Casablanca in Morocco. Her temperature was tested seven times before she flew from Heathrow to Glasgow and she was cleared to travel.
She later became feverish and followed advice given to her at Heathrow to contact local services and was admitted to an isolation facility at the Brownlee unit in Gartnavel Hospital, Glasgow, at 8am on December 29.
After a blood sample tested positive for Ebola, she was transferred in a military plane to the Royal Free Hospital by 8am on December 30.
In a statement last week her relatives said: ''We would like to thank all our friends, family and the members of the public who have contacted us with support following Pauline's diagnosis with Ebola. We have been very touched by the kind words."
They thanked those working at the hospital, adding: "We want to thank all the staff caring for her for their kindness, support and compassion."
Ms Cafferkey's health deteriorated in the new year and on January 3 the hospital announced her condition was critical.
Officials from Public Health England and Health Protection Scotland are reviewing the UK's screening procedures for Ebola after it emerged she had been cleared to fly from London to Glasgow despite her temperature being checked seven times after she landed at Heathrow.
Her diagnosis has brought fresh scrutiny on the UK's preparedness for cases of Ebola, which can only be contracted by coming into contact with the bodily fluids of an infected person.
Previously, any possible Ebola carriers were advised to avoid crowded places and long journeys on public transport within the 21-day potential incubation period once they arrived back in the UK.
Health Secretary Jeremy Hunt said the guidance has since been strengthened to ensure anyone from a high-risk group who feels unwell is reassessed and advice will be sought immediately from an infectious diseases specialist.
UK-based passengers on the flight Mrs Cafferkey was on from Casablanca, Morocco, to Heathrow and those on the flight from London to Glasgow have been contacted, he said.
Scotland's First Minister Nicola Sturgeon has hailed the ''quiet heroism'' of the nurse and others who ''make all of us safer by placing themselves at risk''.
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