The UN health organisation did not agree to the request, saying that instead it will work to give "the best care possible" to the doctor in Sierra Leone, including access to experimental drugs.
Dr Olivet Buck is the fourth doctor from Sierra Leone to come down with Ebola, which has been blamed for 2,400 deaths in West Africa, according to the WHO. Liberia, Sierra Leone and Guinea have recorded the vast majority of cases.
Buck, a citizen of Sierra Leone, would be the first doctor from one of the countries hit hardest by Ebola to receive treatment abroad. The other three doctors from Sierra Leone died in the country.
The letter, sent to the WHO's country representative on Friday and seen yesterday by The Associated Press, said Buck tested positive for Ebola on Tuesday. "We have been informed that Dr. Buck is quite ill," it said.
In an emailed statement, WHO spokesman Tarik Jasarevic said the WHO could not comply with Sierra Leone's request.
"WHO is unable to organise evacuation of this doctor to (Germany) but is exploring all options on how to ensure best care," he said.
There is no licensed treatment for Ebola, though a small number of patients have received unproven treatments, with mixed results. It is not clear how these treatments influenced whether the patients recovered or not.
Because Ebola is only transmitted through contact with the bodily fluids of people showing symptoms or from dead bodies of Ebola victims, health workers have been especially vulnerable as they respond to the worst outbreak in history.
More than 135 health workers have died so far, exacerbating shortages of doctors and nurses in West African countries already hindered by shortages of health workers.
