SAN FRANCISCO -- A toddler who needed heart surgery was in good condition Friday morning, less than a day after her illegal immigrant parents were granted a one-year stay in the United States.

Hazelle Roa, born 17 months ago with a rare chromosomal abnormality, was resting comfortably with her parents by her side at University of California, San Francisco's Medical Center, said UCSF Children's Hospital spokeswoman Phyllis Brown.

Her parents, Victor and Maria Roa, entered the country illegally in 1990 and want to stay in the United States because Hazelle's condition requires specialized care she would likely not be able to get in their native Mexico.

Before U.S. Customs and Immigration officials accepted their request Thursday and granted them a one-year extension, the Roas were under order to leave by July 26.

"The decision was made on the basis of what ICE believes is in the best interest of this family," said Lori Haley, spokeswoman for ICE.

Hazelle was born in the United States and has been monitored by a team of physicians. Her doctors have written letters in support of keeping her family in the United States, saying the child was unlikely to get the medical attention she needs to live a full life elsewhere.

"Her condition is essentially unique," said Stephen Wilson, medical director for the pediatric unit at UCSF, just before the toddler was checked in for a heart operation. "She's really dependent on the technical intervention we're providing here. It's quite critical for her ongoing survival."

Hazelle underwent an exploratory heart procedure Thursday to open up a constricted heart artery -- one of the consequences of her genetic disorder -- and help doctors decide whether she would need further heart surgery.

Although the procedure went well, doctors said Friday they were not able to widen the blood vessel connecting her heart and one of her lungs enough. They had to insert a stent, a rigid tube, to keep the blood flow at normal levels. But because they child is so young, they said she'll have to undergo a diagnostic heart scan every two years, and have the stent substituted as she grows -- about once every four years, Wilson said.

"This has obvious implications for her family's immigration situation," said Wilson. "One year is not necessarily a time frame that will work for this particular problem."

Meanwhile, the Roas want their immigration case to be reopened. They're also asking for permanent residency under a rule that allows undocumented immigrants to remain if their departure would cause extreme hardship to an American citizen.

"This is an extraordinary remedy that's not easily given out, but their profile fits," said the family's attorney, David Lunas. "They're deserving of it."