A second Dallas hospital worker who provided care for the first Ebola patient diagnosed in the U.S. has tested positive for the disease, pointing to lapses beyond how one individual may have donned and removed personal protective garb.

It's not clear how the second worker contracted the virus. Authorities declined to say what position she holds at the hospital or the type of care she provided to Thomas Eric Duncan, who was diagnosed with Ebola after coming to the U.S. from Liberia. Duncan died Oct. 8.

Officials have said they also don't know how the first health worker, a nurse, became infected. But the federal Centers for Disease Control and Prevention said "an additional health care worker testing positive for Ebola is a serious concern."

"What happened there (in Dallas), regardless of the reason, is not acceptable. It shouldn't have happened," Anthony Fauci, director of the Institute of Allergy and Infectious Diseases of NIH, said on MSNBC Wednesday.

The worker at Texas Health Presbyterian Hospital was monitoring herself for symptoms, Dallas County Judge Clay Jenkins said. The unidentified woman reported a fever Tuesday. She was in isolation within 90 minutes, Jenkins said.

"We are looking at every element of our personal protection equipment and infection control in the hospital," said Dr. Daniel Varga, chief clinical officer for Texas Health Resources, which operates Texas Health Presbyterian Hospital Dallas.

Dr. Tom Frieden, head of the CDC, has acknowledged that the government wasn't aggressive enough in managing Ebola and containing the virus as it spread from an infected patient to a nurse at a Dallas hospital.

"We could've sent a more robust hospital infection control team and been more hands-on with the hospital from day one about exactly how this should be managed," he said Tuesday.

The second case may help health officials determine where the infection control breach is occurring and make practices safer for health workers everywhere. For example, if both health workers were involved in drawing Duncan's blood, placing an intravenous line or suctioning mucus when Duncan was on a breathing machine, that would be recognized as a particularly high-risk activity. It also might reveal which body fluids pose the greatest risk.

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