A murderer who brutally killed his partner at their home in Adelaide's north has finally confessed to also murdering her two children.

Steven Graham Peet, 32, strangled his partner Adeline Yvette Wilson-Rigney and her children to death in May 2016.

Their bodies were found bound and gagged and wrapped in sheets and clothing in various locations around the rental property at Hillier.

Adeline Yvette Rigney-Wilson was murdered by her partner. (Facebook)

Korey and Amber's bodies were found at a Hillier property in 2016. (Facebook)

Triple murderer Steven Peet. (9NEWS)

At trial last year, he admitted murdering Ms Wilson-Rigney, 28, but denied killing six-year-old Amber Rigney and five-year-old Korey Mitchell.

But the trial was derailed when Peet's defence team produced a psychologist report which claimed he was in a "dissociative state" when he killed the children, and therefore was not guilty of murder.

The court today heard the psychologist has since changed his opinion after being presented with new evidence from prosecutors, stripping Peet of his mental incompetence defence.

Peet closed his eyes and bowed his head as he twice muttered "guilty" when he was asked how he pleaded to each murder charge.

The admissions triggered shrieks of disbelief from the victims' family members, who were sitting in the public gallery.

Stephen Peet today admitted killing the children.

A court heard horrific details of the children's murders.

Police at the scene of the triple murders. (9NEWS)

The children's grandfather, Steven Egberts, and his partner Janet Wells wept uncontrollably.

Outside court, Ms Wells said there were "no words" to describe how she felt about their killer.

When asked if she was relieved that Peet has pleaded guilty, she answered "it's about time".

Ms Wilson-Rigney's mother Donna Rigney was too emotional to speak on camera but the Victims' Rights Commissioner spoke on her behalf.

"She is glad that this man has accepted responsibility for the crimes that he committed," Michael O'Connell said.

"This is actually the end of a very distressing phase for her but also that she will now have a life long sentence - a life long without her grandchildren and without her daughter."

Ms Wilson-Rigney was murdered during an argument with Peet over their relationship.

Prosecutors said Peet was "infatuated" by the mother, but that she had recently decided to leave him.

Peet will be handed a mandatory life jail sentence for the three murders.

The judge is yet to determine his non-parole period.

The victims' family members will tell the court how the crimes have affected them during a pre-sentencing hearing on Friday.