The university said it was "highly likely that a non-residential member of our community has contracted the Covid-19 virus from outside the university" and that the campus will be closed until Friday, which was due to be the last day of the spring term.
The university is hoping to reopen after the Easter break on April 13 but in the meantime students are being taught online and people staying in halls of residence have been told to return to their permanent homes.
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A statement from the centre said: "University Centre Shrewsbury has closed with immediate effect from today and has recommended students who live in student accommodation to return home if possible.
"The university was informed on Sunday, March 15, that it was highly likely that a member of our UCS community may have contracted COVID-19.
"This person does not live in student accommodation, and is now in self-isolation under the advice of local health care professionals.
"Students will be taught and supported online until term ends this Friday, an approach for which UCS has had recent experience after the flooding in Shrewsbury and which has been adopted by universities across the country in the light of the current position with coronavirus.
"This is a rapidly changing situation nationally and University Centre Shrewsbury, with support and guidance from the University of Chester, has been taking all necessary precautions to protect its students, staff and guests.
"Last week, all forthcoming external events were cancelled and earlier measures taken by UCS included suspending all staff travel and cancelling external meetings."
Professor Anna Sutton, the university centre's provost, said keeping students, staff and visitors safe was the primary concern.
“We will take all measures necessary to ensure their well-being during this extraordinary and unprecedented time," she said.
“Our thoughts at this time are with the affected person and their family. We wish them a swift and full recovery.”
Meanwhile Hodnet Primary School near Market Drayton has closed for a deep clean after a student showed symptoms of coronavirus last week.
Current government guidance is for schools and colleges to stay open unless somebody has contracted coronavirus.
As of Sunday 35 people had died in the UK after testing positive for Covid-19, including three patients in Wolverhampton.
No deaths have been confirmed in the Shropshire, Powys or Telford & Wrekin council areas so far.