Public Health Wales confirmed one person had been infected abroad and the two patients were from the same household.
Both patients were admitted to a specialist hospital, but one has since been released.
Health Secretary Matt Hancock confirmed the outbreak in Wales on Thursday.
He told MPs: “The tracing and isolation system was essentially built for very important but very small outbreaks.
“As Health Secretary you’re dealing with these sorts of outbreaks all of the time.
“I’m currently dealing with a monkeypox outbreak and cases of drug-resistant TB and that is absolutely standard and the lack of that capability at the start meant that the options that we had were fewer.”