DALLAS — A nurse here became the first person to contract Ebola within the United States, prompting local, state and federal officials who had settled into a choreographed response to scramble on Sunday to solve the mystery of how she became infected, despite wearing protective gear, and to monitor additional people possibly at risk.

The news further stoked fears of health care workers across the country, many of whom have grown increasingly anxious about having to handle Ebola cases. The confirmation on Sunday of the second Ebola case in Dallas — four days after the death on Wednesday of the first patient, Thomas Eric Duncan, 42, a Liberian who arrived in this country in September — opened a new and more frightening chapter in the unfolding public health drama.

While the new Ebola patient was not publicly identified, officials said that she was a nurse who had helped treat Mr. Duncan at a hospital here and that she may have violated safety protocols. It was the first confirmed instance of Ebola being transmitted in this country. Officials expanded the pool of people they had been monitoring, because the nurse had not been among the 48 health care workers, relatives of Mr. Duncan and others whom they were evaluating daily.