Anni Manukyan and her husband Ashot, from Los Angeles, had two embryos implanted last year but to their dismay the procedure wasn't successful.
As they came to terms with their disappointment, the couple say they were asked to return to CHA Fertility Center to for what they believed was a routine appointment.
There, their cheeks were swabbed.
A day later they claim they were asked to go a back to the clinic where they were told that the swab had been a DNA test, and that it had confirmed that they were the biological parents of a baby boy who had been born to another couple in a shocking blunder.
The baby was one of twin boys born to a couple identified only as Y.Z. and A.P. from New York, who are also suing the clinic after tests allegedly showed neither boy was related to them or to each other.
In May, Anni and Ashot met their baby, Alec, then six weeks old, in a hotel lobby.
The couple are represented by lawyer Adam Wolf from law firm Peiffer Wolf Carr & Kane and have spoken out about how the alleged mix-up has left them heartbroken.
Anni said, 'Who wants to meet their child in the lobby of a hotel? It was heartbreaking, it was horrible.'
She also expressed fears that another embryo belonging to her had been given to another stranger, as it was claimed the embryos implanted in her were not actually her own.
'I'm just praying to God that I don't have another son or daughter out there,' Anni said.
The couple are also thankful to the New York woman who carried and delivered baby Alec.
She said, 'I pray for her every day. She was a victim of this as much as I am. She's a lovely lady. She raised my baby inside of her and after he was born.'
The biological parents of Alec's 'twin', who have also been reunited with their son, have not come forward.
Mr Wolf said: 'This incredible series of events demonstrates CHA's shocking incompetence. While I have handled hundreds of cases of fertility-center misconduct, this tragedy at CHA is among the most egregious I have seen.'
The clinic has yet to publicly comment on the lawsuits.