Chief Public Health Officer Nicola Spurrier said 16 of the new cases were linked to one event at suburban Norwood last weekend.
The other two involved travellers from interstate.
All cases who attended the Norwood event were fully vaccinated. Former SA premier Jay Weatherill was among them and has tested positive.
The cluster has forced 210 close contacts to quarantine for seven days but officials expect that number to increase.
South Australia Chief Public Health Officer Professor Nicola Spurrier (pictured) believes the cases are linked to someone who entered the state from NSW
"It's quite clear we do have COVID in our state now," Professor Spurrier said.
"This looks as though it was linked to someone coming from NSW. But I think there will be other chains of transmission."
"This disease is a real leveller. It can affect anyone."
Professor Spurrier said SA was now in a "transitional" stage as she urged everyone to follow COVID-19 restrictions and protections, including