'Baby Girl' Veronica Has Not Been Returned To Adoptive Parents Who Won Custody (VIDEO)

WATCH: 'Baby Girl' Veronica's Adoptive Parents Speak Out

As the custody battle between a South Carolina couple and the biological father of "Baby Girl" Veronica rages on, the 3-year-old's adoptive parents, Melanie and Matt Capobianco, have spoken out about the heartache of losing their daughter.

The Capobiancos appeared on NBC's "Today" show Friday morning to explain how the child's birth father, Dusten Brown, allegedly refuses to return "Baby Girl" -- as she is referred to in court documents -- to their custody, even though the court recently ruled in their favor.

"Nobody showed. They could have asked for a different time, a different date. But, none of his attorneys did in court, so we assumed that he was going to show," the girl's adoptive father told journalist Savannah Guthrie.

For those unfamiliar with the story, the case revolves around the custody of a young Cherokee girl who was adopted by the Capobiancos in 2009 but was handed over to Brown in 2011 after a South Carolina ruled he was favored as the child's custodian. Brown originally signed over his parental rights to the girl's mother while she was pregnant before he deployed to Iraq. He then sought to custody once he learned the mother, who was already caring for two other children, had put his daughter up for adoption.

Since Brown is a member of a federally recognized Indian tribe, the state's Supreme Court upheld the lower court's ruling, citing the 1978 Indian Child Welfare Act. The Capobiancos were then forced to turn over the daughter they had named Veronica -- and raised for two and a half years -- to her biological father.

"It was the worst day of our lives, but we also knew then and we know now that it isn't about the adults. It's about her," Veronica's adoptive mother told The Associated Press Wednesday. "We had to follow the rule of law. And if we were going to have a chance to get her back, we had to hand her over."

But the South Carolina family did not give up and pursued case to the nation's highest court, winning back custody of "Baby Girl" Veronica earlier this year after the U.S. Supreme Court ruled that the Indian Child Welfare Act did not apply. A South Carolina Family Court judge then granted full legal custody to the Capobiancos and finalized the adoption last week so the toddler could be returned to the family.

Though Brown filed a petition asking the Supreme Court to block the court-ordered handover, his legal request was denied.

However, the Capobiancos say Brown and his family are refusing to return the girl while he's away serving a short term with the National Guard this month.

"They cut off all communication. We never got to see her, talk to her, she didn't know what had happened to us," Melanie Capobianco explained. "We obviously want it to go as smooth as possible. We don't want any trauma."

South Carolina Judge Daniel Martin has since ordered state and federal prosecutors to locate "Baby Girl" Veronica and deliver her to her adoptive parents.

See a clip from Melanie and Matt Capobianco's "Today" show interview above, or watch the full report in the video below.

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