Staff at a Monash IVF laboratory in southeast Melbourne incorrectly transferred a woman's own embryo to her on Thursday instead of that of her partner, as the couple had requested.
The duo are believed to be in a same-sex relationship.
The company, which is based in Melbourne but has clinics around Australia, apologised to the pair and launched an internal investigation.
But the Victorian Health Regulator will independently probe Monash IVF and how the error occurred.
"Families should have confidence that the treatment they are receiving is done to the highest standard," state health minister Mary-Anne Thomas said.
"It is clear Monash IVF has failed in delivering that - which is completely unacceptable."
Ms Thomas said Monash IVF was required to co-operate with the investigation and provide clear answers.
In a notice to the stock market, the company said it would set up additional verification processes and patient confirmation safeguards.
"Whilst industry-leading electronic witness systems have and are being rolled out across Monash IVF, there remain instances and circumstances whereby manual witnessing is required," it said.
The Reproductive Technology Accreditation Committee certifying body and insurers were also informed of the error.
Monash IVF expects the mix-up to fall within its insurance coverage.
Its profit guidance remains unchanged, but the news sent the company's share price tumbling by more than 23 per cent as of 1pm AEST.
Monash IVF revealed in April that a woman at a Brisbane facility had another patient's embryo incorrectly transferred to her due to "human error".
The mistake was picked up in February after the birth parents asked for their remaining embryos to be transferred elsewhere and an extra embryo was found in storage.
Monash IVF apologised, saying it was confident it was an isolated incident.
An independent review led by leading barrister Fiona McLeod has been expanded following the second transfer mix-up, with findings to be released "in due course".
Associate Professor Alex Polyakov, a fertility specialist at Melbourne's Royal Women's Hospital, said the latest incident was also likely due to human error.
He said the mistake in Melbourne was equally serious to the one in Brisbane.
"I do think it's a little bit easier to make this mistake because the partners are linked on the system," the Genea Fertility Melbourne medical director told AAP.
"When they took the embryo, they check the partner's name, they check the patient's name and in this particular incident obviously there was a bit of a mix-up because it was part of the same couple.
"Embryos belong to a couple. It doesn't belong to a person. They took the wrong embryo but it still belonged to this couple."
About one in 18 babies is born via IVF in Australia.
Prof Polyakov said the two errors were the first he was aware of occurring in Australia since the IVF industry began operating 40 years ago.
He called for more vigilance and extra layers of protection from clinics but conceded no system could be made entirely foolproof.
"These things are like aeroplane crashes - they happen very, very rarely," he said.
"The processes that we use in our lab and every other lab in Australia are probably the best in the world in terms of preventing these incidents, and it's really just terrible bad luck for Monash.
"I would call it a black swan event where there are a number of errors that have to take place for this to occur."
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