The cafe has been shut since Sunday (June 6) and will "remain closed until further notice", say Preston City Council.
It has not confirmed exactly how many staff have tested positive but said all employees are self-isolating at home.
The cafe is due to host a wedding on Saturday (June 12) and the Council has confirmed that this will go ahead following a deep clean of the venue.
It said the public toilets are still open but will close earlier than usual for cleaning.
The Council said it does not know at this stage when the cafe is likely to reopen.
Coun Robert Boswell said: "Unfortunately a couple of our café staff have tested positive for Covid and, in line with current procedures, the remaining staff are self-isolating and will take further tests before returning to work. We wish them a full and speedy recovery.
"The Pavilion has been deep cleaned and will be safe and ready for a private function this weekend. The café remains closed until further notice."
It is the latest city venue to close this week due to Covid-19, as the latest figures show 13 out of 17 Preston neighbourhoods saw an increase in infections over the last 7 days.
A total of 19 staff members were told to self-isolate at home and four employees had tested positive. But the pub said it would remain open.
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