DALLAS — Some nurses donned layer after layer of protective garb but unknowingly raised their risk of exposure to the Ebola virus when taking the gear off. Some wore gowns that left their necks uncovered and haphazardly applied surgical tape to the bare spots. And it was two days after the Ebola victim Thomas Eric Duncan was admitted before personnel began wearing biohazard suits.

For Texas Health Presbyterian Hospital, the last two and a half weeks have been a nightmare without end. And with the announcement early Wednesday that a second nurse who cared for Mr. Duncan had been infected, scrutiny of the hospital intensified as officials sought to calm both workers and patients.

Long regarded as one of the finest hospitals in Texas, Presbyterian has faced continuing criticism — first for its initial misdiagnosis of Mr. Duncan, which delayed his care and placed others at risk; then for issuing contradictory statements about why its doctors did not suspect Ebola; and now for failures in safety protocol that led to the infections of two of its own. If the hospital has served as a canary in a coal mine for the country’s Ebola response, the results have not inspired confidence.

Dr. Thomas R. Frieden, director of the Centers for Disease Control and Prevention, who has received his own measure of criticism for the C.D.C.’s response, seemed to acknowledge as much on Wednesday when he described some of the hospital’s lapses. He also announced that the second nurse would be flown to Emory University Hospital in Atlanta, which has successfully treated two American Ebola patients in one of the country’s four special biocontainment units.