The staff member worked at St Ann's aged care facility on Christmas Day and was tested for COVID-19 on Monday.
They returned a positive result on Tuesday and the Public Health Services and the Department of Health's aged care emergency operations centre is now working with the facility.
Management at St Ann's advised the government that staff wear masks during their shifts.
Residents' family members are being contacted to inform them of the situation.
Anyone who was at the St Ann's aged care facility in Davey St, Hobart, between 7am and 2pm on Christmas Day should monitor themselves for symptoms and get tested immediately if any develop, even if mild.
Tasmania now has 300 active infections, the majority of which, 158, are being managed at home.
There are 63 people in community management clinics and one person is in hospital for an unrelated medical condition.
A further 23 people are still being assessed.
Tasmania was free of COVID-19 when it reopened to fully vaccinated travellers from mainland hotspots on December 15.
Currently the state requires travellers to return a negative PCR test in the 72 hours before travel into Tasmania.
However, with testing sites in some states breaking under the workload, Tasmania is considering if that requirement can be replaced with a rapid antigen test.
The topic is expected to be discussed at National Cabinet on Thursday.
National Cabinet will meet to discuss a number of matters, including the definition of close contacts, testing regimes, and supply.
Work is also being finalised on Tasmania's vaccination roll-out for five-to-11 year-olds and the booster program roll-out.
"I expect to be able to provide an update on these matters to Tasmanians post tomorrow's National Cabinet meeting," Premier Peter Gutwein said in a statement on Wednesday.
"Tasmanians are encouraged to continue to use common sense: wear your mask indoors, practice good hygiene, get tested if unwell even with mild symptoms, and use the Check in Tas App.
"Importantly, if you're unwell even with the mildest of symptoms, get a test and isolate until you receive the result."
The Tasmanian opposition have called for better testing capabilities in regional and rural parts of the state.
"Now that Premier Peter Gutwein has opened our borders, he either needs to ensure testing facilities can be stood up throughout the state, or explain why testing is not being provided across regional Tasmania," acting Labor leader Anita Dow said in a statement.
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