Claire Roberts, who was nine, died in 1996 and Adam Strain, who was four, died in 1995 at the Royal Belfast Hospital for Sick Children.
On Thursday, a lawyer for the trust told the hyponatremia inquiry that it was liable for the deaths - caused by how their fluids were managed.
The Belfast Trust made a full and frank admission of liability for the death of Claire Roberts.
At the time, her parents say were told she had died from a stomach virus which spread to her brain, but now almost 17 years on the inquiry heard an apology from the Belfast Trust to the Roberts family.
Claire's mother Jennifer told UTV: "We brought Claire to the Royal Belfast Hospital for Sick Children and we left her in their care.
"We left Claire on Tuesday, 22 October 1996 thinking she was sleeping and we said to the nurses that evening, 'We'll see you in the morning.'
"And then we were wakened at 3.45am, in the early hours of 23 October 1996, to say that we had to go up to the hospital that there was difficulties with Claire."
The child had died.
They failed Claire and I have to add they failed us as parents.
It was not until 2004 and a UTV documentary 'Insight: When Hospitals Kill' that the Roberts began to ask questions.
Prior to that, they had never heard of the word hyponatremia.
Claire's father Alan told UTV:"We realised that we had to go back and ask additional, further questions about Claire's treatment and we had to ask questions about fluid, fluid management.
"And this word hyponatremia was a new word to us, we had never heard that word ever mentioned before."
On Thursday, following the Trust's admission, Alan said they welcomed the apology, admitting that it had been "a very difficult 17 years."
He said that it had taken 17 years - including "two really horrendous years" of going through a public inquiry.
He continued: "This should never happen to any family.
"No family should have to go through what we have been put through, we've lost a daughter.
"It's bad enough, the worst life experience anyone can have is a loss of a child.
"People who have lost children will know what I'm talking about and that's the worst catastrophe any parent can have."
Jennifer added: "Claire was much loved by us - her mum and dad, her brothers, her grandparents, her family.
"She was such a happy, active young girl. She brought us so much love.
"This process has robbed us of the happy memories of Claire, of being able to talk about her in happy terms. It has re-triggered grief and has brought out an awful lot of anger."
The trust also made a full admission of liability for the death of Adam Strain in 1995.
We do not underestimate how difficult this must be for their families and we would like to extend a sincere apology to them.
In a statement, the trust said it "fully acknowledges the shortcomings in care and treatment that both children received in the former Royal Hospitals Trust".
The lawyer also read a statement for the Southern Trust which acknowledged that guidelines were not properly implemented at Craigavon Area Hospital where a third child, Conor Mitchell, died in 2003 when he was 15.
His family have welcomed an apology from the Southern Trust.
They said: "The family of Conor Mitchell are extremely pleased that the Southern Trust has finally accepted the failure to provide Conor with the level of care he and they would have expected.
"Despite the reservation to the admission, the full apology is a welcomed turn of events. The fact that they have had to fight for ten years to receive this remains a matter of regret.
"The family hope that these admissions will signal a new era of openness and transparency for the NHS in dealing with future traumatic events."
The inquiry was set up to look into the deaths of five children and is set to run for around another five weeks - with Justice John O'Hara expected to publish his findings early next year.
Earlier this year, the Western Trust admitted liability in the case of Raychel Ferguson's death in 2001.