The baby at the center of is now in his parents' custody and receiving care, their lawyer said Thursday.

The child, who was born Wednesday morning in Texas, has hypoplastic left heart syndrome, a rare heart defect that requires a series of surgeries, the first typically happening in the first weeks of life.

When they learned of the diagnosis months ago, the parents, identified in court documents as Nausheen Gilkar and Omar Ahmed of California, requested that the surrogate, McKenna West of Alaska, get an abortion, according to Texas Attorney General Ken Paxton, who intervened in the case this week.

West refused and went to Texas before the baby was born, prompting Paxton to secure a court order saying the baby must receive lifesaving medical care after birth. The court also ruled that Gilkar and Ahmed, not West, are responsible for medical decisions within the requirements of the court order.

"Our clients' son was born yesterday morning and he is now in their physical custody while he receives expert medical care at a tertiary care center," attorney Lee Budner, who represents Gilkar and Ahmed, said in a statement. "Thankfully, McKenna West's baseless attempt to interfere with his medical care has been soundly rejected by the courts. Right now, our clients are continuing to place their baby's health and well-being first, following the advice of the baby's medical team, as they always have and just as any loving parent would."

"As if the heartbreak of his condition were not enough, they are devastated to see their family tragedy transformed into political theater by the Texas Attorney General's office and McKenna West," Budner's statement continued. "Our clients' only focus at this time is ensuring that their baby receives the medical care that he vitally needs and that they, as his parents, are able to spend every precious moment they can with their beautiful newborn son."

Due to the court orders, West was not allowed any contact with the child after his birth, her attorney, Lincoln Davis Wilson, told CBS News Wednesday. Wilson said West intended to continue efforts to gain custody of the child.

"We are working on challenging those orders of parentage because we think that Texas law makes a woman who gives birth the child, the mother of the child," he said.

Wilson also said that aside from the known heart condition, signs were positive for the baby, who is called Gabriel in the court documents. Wilson could not comment on where the child was being treated, but said he was receiving care "at a facility that has an excellent track record of success in performing it."