The BBC named the patient as Pauline Cafferkey, and local media described her as nurse with 16 years' experience who normally worked at a Scottish health centre.
Sturgeon said she could not confirm the patient's identity for confidentiality reasons.
The London hospital's 'High-level isolation unit' will allow doctors to treat Cafferkey while she lies in a plastic tent, limiting the scope for the disease, which is transmitted by contact with bodily fluids, to be passed to medical staff.
The World Health Organization said on Monday that the number of people infected by Ebola in the three West African countries worst affected by the outbreak -- Sierra Leone, Liberia and Guinea -- had passed 20,000, with more than 7,842 deaths so far.
Read MoreIf you think Ebola was bad...
Cafferkey, a National Health Service worker who had been working in West Africa with the charity Save the Children, flew from Sierra Leone to Glasgow late on Sunday on a British Airways flight via Casablanca in Morocco and London's Heathrow.
She was diagnosed with the virus on Monday after developing symptoms overnight and was initially treated at a Scottish hospital.
Local media said she is in her late 30s and had been working in the Kerry Town Ebola treatment centre outside Sierra Leone's capital, Freetown.
A Reuters journalist who visited the centre last week said it was a huge, pristine facility built by British army engineers, with around 80 beds and stringent cleansing procedures far superior to those of a nearby Sierra Leonean treatment centre.
With paths laid with new gravel and whitewashed walls, the centre even had a gazebo in the "red zone" for recovering patients to sit and chat. Child patients were given a cuddly toy on arrival. On discharge, the toys would have to stay inside the red zone, so each child was given a new, ebola-free toy.
At every doorway there was a barrel of disinfectant. Health workers preparing to enter the red zone all had spotters to check equipment as they changed into protective clothing.
Read MoreAnd Time's 2014 Person of the Year is...