The Cross Guns in Pant, near Oswestry, says it could be a fortnight before it re-opens.
Bar manager Andrew Keegan said the pub, run by mother and son Maureen Hayle and Nigel Hayward, had a Track and Trace system in place.
"We have the app code and a book for customers to sign," he said.
"After one of our staff lost their sense of taste and smell they went for a Covid-19 test immediately and that has come back positive.
"We have closed the pub and informed the NHS."
"It is likely that we will be closed for the next two weeks."
The pub was taken over by Maureen and Nigel in February this year, just weeks before the national Covid-19 lockdown.
Mr Keegan added: "We have built up a wonderful community following of loyal customers and we are very keen to be open again as soon as possible."