Mr. Duncan, a Liberian who arrived in America last month, was initially sent home from the hospital although he had a fever and reported coming from Africa. His case was diagnosed on Sept. 30, two days after he was admitted and put in isolation. At some point, Ms. Pham contracted Ebola after having extensive interactions with him. The hospital said she was in “good condition,” and for the first time on Tuesday, Ms. Pham made a public statement, as the hospital said the costs of her care would be covered and would not be a financial burden on her or her family.

“I’m doing well and want to thank everyone for their kind wishes and prayers,” she said.

At the same time, Dr. Frieden said that the 48 people whom officials started monitoring two weeks ago had passed a critical period and were now unlikely to contract the disease. They were more than 14 days into the 21-day maximum incubation period.

One of them is Mr. Duncan’s fiancée, Louise Troh, 54, who has been under state-ordered quarantine with her 13-year-old son and two other young men. “Right now, we’re O.K.,” she said in a phone interview from the house where she and the others have been staying, in an undisclosed location. “I’m healthy. Everybody’s fine. God is in control.”

The Rev. George Mason, Ms. Troh’s pastor at Wilshire Baptist Church, drove on Tuesday to pick up an application for a new apartment for Ms. Troh. He said she would be starting over with almost nothing as her old apartment was shut by officials and decontaminated.