One rare heart defect is Tetraology of Fallot. It occurs in about five out of every 1,000 births, according to my research. While this defect is usually quickly diagnosed, and can be treated with surgery, sometimes the complete surgery is delayed until the baby is stronger. To that end, a surgeon sometimes places a shunt in the heart to improve blood flow so the baby can grow and improve.
This is a complex disorder and it's a testament to how far modern medicine has come in the past 60 years that, in say, 1955, a child born with this disorder was considered terminal. Most children with this disorder did not live to their second birthday. Now, it can be taken care of surgically, and most patients can lead a normal life.