The 26-year-old man, interviewed by the Vice News Austrian website and identified only as "Valentin", said that he was stopped last Tuesday.
Doctors had told him to wear his mask to protect his immune system, which had been so severely weakened by a intensive course of chemotherapy and other treatment that he initially had to spend eight weeks in isolation.
"The first thing they asked me was whether I spoke German or whether I was foreign," Valentin told Vice News, which specializes in stories which might not get into mainstream media. "When I explained that I was from Vienna, things calmed down a bit. But then they went after my face mask."
The police explained to him that he was contravening the law, which carries an 80 euro ($98) fine.
"I had asked my doctor about exactly this. But at the time he just laughed and said it would be obvious to anyone why I was wearing it".
He proceeded to show the police his medication and searched for his blood test results on his phone.
"The whole thing was quite a stressful experience, but in the end they believed me," Valentin said.
He was warned that next time he would need a doctor's certificate or would face a fine. Police spokesman Patrick Maierhofer admitted the man was stopped and said "the matter was sorted out after a short conversation".
"Covering your face for medical reasons is an exception. In such cases the police must be given credible proof that it is being worn for medical reasons," Maierhofer told the APA agency.
The law, which came into force on October 1st, bans the full Islamic veil, known as the burqa or the niqab, in public places. But in order to avoid being sued for discrimination, the government outlawed at the same time any item of clothing that covers the face.
The ban on the full-face veil, which remains a rare sight in Austria, was seen as an effort by the two parties then in government, the centre left Social Democrats (SPOe) and centre-right People's Party (OeVP), to halt a rise in support for the anti-immigration Freedom Party (FPOe).
However, the FPOe went on to gain seats in October's elections and is now the junior partner in an OeVP-led coalition government, with the SPOe going into opposition.