On Sunday, the British aid worker flew back to Glasgow from Sierra Leone, where she was treating patients with ebola with Save the Children.
She was placed in an isolation unit on the Monday morning when she began showing symptoms.
She's now been named as nurse Pauline Cafferkey.
This morning, she was flown to the Royal Free Hospital in London for specialist treatment in a military plane.
The Royal Free Hospital is the same hospital where British aid worker William Pooley was treated.
Most Rev Welby said via Twitter: "The Ebola of the Glasgow Health worker shows the courage of those serving in S Leone, we owe them thanks and respect, let us pray for her."
The Scottish government's confirmed a second person is being tested for the ebola virus. Meanwhile, another person is being tested in Cornwall.
The ebola virus has killed nearly 8000 people in the West African countries of Liberia, Sierra Leone and Guinea, according to the World Health Organisation.
SCIAF Director Alistair Dutton said: "Our thoughts and prayers are with Pauline Cafferkey and all those who are putting themselves in harm's way in West Africa to care for those with Ebola.
"Ms Cafferkey's case highlights that the Ebola epidemic remains a major emergency and needs our urgent attention. I urge the public to donate whatever they can so that SCIAF and our Caritas partners on the ground can help to prevent this deadly disease spreading further."
