The as-yet unnamed woman had been admitted to the hospital on March 2 but died four days later.
It is understood that the patient had been living in Wigton for the past couple of years, but had been brought up in the Maryport area.
Police investigations into the circumstances surrounding her death are in their early stages.
A spokeswoman for North Cumbria University Hospitals NHS Trust, which runs the hospital, said: “We are deeply saddened by the death of a 21-year-old patient and we wish to express our sympathy to the patient’s family.
“We cannot comment further about the circumstances surrounding the patient’s death due to the current police investigation.”
Cumbria Police has confirmed an investigation is underway.
“Police are investigating the circumstances of a death of a 21-year-old woman,” a spokesman said.
“The woman was pronounced dead on March 6 March 2016 after being admitted to the Cumberland Infirmary on March 2 due to illness. The investigation is in the early stages and inquiries are ongoing.”
It is the second death at the infirmary currently being looked into by police, as an investigation is still underway into how Amanda Coulthard died.
The 57-year-old mother-of-two, who lived in Hunter Lane, Penrith, is understood to have been at the infirmary for a routine procedure last April, but subsequently died.
The hospital trust was one of 11 plunged into special measures in July 2013, following a review by Sir Bruce Keogh into hospitals with shocking mortality ratios.
Improvements have been made in recent years, but the health watchdog the Care Quality Commission says it still requires improvement.
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