(Adds patient discharged from Maryland hospital)

By Lisa Maria Garza and Richard Valdmanis

DALLAS, Oct 7 (Reuters) - The Ebola patient fighting for his life in a Dallas hospital is on a ventilator and a kidney dialysis machine to help stabilize his health, the hospital said on Tuesday.

Liberian national Thomas Eric Duncan, the first person diagnosed with the deadly virus on U.S. soil, has also been given the experimental medication brincidofovir. A hospital in Nebraska said it is using the same drug to treat an American journalist who was airlifted from Liberia and arrived Monday.

Texas Health Presbyterian Hospital said in a statement that Duncan's liver function declined over the weekend. It said although it has since improved, "doctors caution that this could vary in coming days."

Duncan is being sedated. His temperature is normal and his diarrhea has slowed, according to family members who spoke with his doctors at the hospital on Tuesday. They did not see Duncan, who has been in an isolation unit since Sept. 28.

Authorities in the United States and the public are on alert following Duncan's diagnosis more than a week ago, which raised concerns that the worst epidemic of Ebola on record could spread from three hardest-hit impoverished countries - Guinea, Sierra Leone and Liberia. Duncan, who arrived in Texas in late September on a commercial flight from Liberia, has been in critical condition since Saturday.

The first case of Ebola being contracted outside of West Africa was reported in Spain and the World Health Organization expects more cases in Europe.

Freelance NBC cameraman Ashoka Mukpo is being treated at the Nebraska Medical Center, which cared for one other U.S. national flown out of West Africa after contracting Ebola and was later discharged.

A Maryland hospital said late Tuesday it discharged a patient exposed to Ebola in Sierra Leone. He was flown to the United States and admitted to the National Institutes of Health Clinical Center in Bethesda, Maryland, on Sept. 28. The patient, who was not named, will remain in his home for 21 days since his exposure to the virus, a needle stick injury.

The drug used in Dallas and Nebraska, brincidofovir, was developed by Durham, North Carolina-based Chimerix Inc. The company said it has been tested in more than 1,000 patients without raising safety concerns.

"We decided this was currently our best option for treatment," said Phil Smith, medical director of the Nebraska Medical Center's Biocontainment Unit, which consulted with U.S. health and drug officials before making its decision.