Genetic counselors in Boston are offering new parents a controversial peek at their baby's future health. It's part of a landmark study that could lead to gene scans for all infants at birth.

By law, every newborn in America gets a blood test for about 30 conditions including cystic fibrosis and sickle cell anemia. A trial underway at Brigham and Women's Hospital uses genomic sequencing to screen for about 1,800 conditions, including some cancers.

By testing babies long before they show symptoms, doctors hope it may be possible in some cases to start treatment early. That could save lives and prevent suffering, reports CBS News correspondent Tony Dokoupil. But it also raises questions of privacy and discrimination, and that's a major reason why most families are still saying "no thanks."

The day after their daughter, Cora, was born, Lauren Stetson and her husband Kyle got an unexpected visit from a genetic counselor.

"I was in full recovery mode, as in, 'I don't care about anything.' I'm just trying to keep our baby alive and recover myself," Stetson said.

The visitor offered a free DNA scan for Cora, a scan that the Stetsons learned could reveal disease-causing variations in their daughter's genetic code.

"I was just trying to make it through the day. So that was definitely something that was a little shocking," Stetson said.

Baby Cora is now one of the first healthy kids in America to have had her genome searched for hidden problems. Doctors found a partial biotinidase deficiency, something that Cora showed no outward signs of having. Had it not been detected, it could have caused a permanent drop in her IQ.

"It's a pretty insane thing to think about," Stetson said.

"She would have still been considered normal. Nobody would have identified her as diseased," said Dr. Robert Green, medical geneticist at Harvard. He's also the co-director of the BabySeq Project along with Alan Beggs, which enrolled Cora and is now recruiting hundreds of other families. Green said you could "absolutely" potentially save a child's life through the scan.

But Green is also warning families about the risks including breaches of privacy and genetic discrimination.