The spokesman said Public Health England has advised university officials that the risk of infection being passed on campus is "low".
He said: "Public Health England (PHE) have informed us that one of the two individuals to have tested positive for coronavirus is a student at the University of York.
"We understand this development will cause concern and anxiety among our students, staff, and the wider community.
"PHE has advised us that the risk of infection being passed to others on campus is low. Current information from PHE suggests that the student did not come into contact with anybody on campus whilst they had symptoms, but investigations are ongoing to fully establish this.
"Our immediate concerns are for the affected student and family, along with the health and continued wellbeing of our staff, students and visitors."
The spokesman confirmed the University of York continues to operate normally.
He added: "We are monitoring the situation closely and we continue to provide as much advice, care and support as we can to our University community.
"If people have any concerns about their health in relation to suspected coronavirus we ask that they follow current PHE advice and contact NHS 111.
"The University has set up a call centre over the weekend for anyone who has any further concerns or inquiries. The telephone number is 01904 809571."