I think you may be referring to congenital analgesia?
Wikipedia defines this as:
Congenital insensitivity to pain (CIP), also known as congenital analgesia, is one or more rare conditions where a person cannot feel (and has never felt) physical pain. The conditions described here are separate from the HSAN group of disorders, which have more specific signs and etiology.
The girl you reference is mentioned in Pain: the science of suffering By Patrick David Wall:
One such case was studied over a period of twenty years up to the time she was a student at McGill University in Montreal. A strong pinch to the foot failed to produce a withdrawal or to provoke pain. When pinched and asked what she felt, she replied in a calm way that "it feels like a very strong pressure, and I know that if you pinch much harder you will injure my foot"
All her other body sensations-touch, pressure, warm, cold, and movement-appeared completely normal.
Unusual accidents do occur in such people in novel situations.
For example, as a child in the deep Canadian winter, she climbed up to look out of the window and knelt on a hot radiator, one could still see line scars on her knees as an adult.
There's also a very interesting article from HowStuffWorks which refers to this same girl:
Congenital analgesia is a rare genetic disorder where the individual is unable to feel pain. You might think this sounds like a good thing, but it's actually a life-threatening condition. Pain serves as a warning against injury, so people who don't feel it can be severly injured or hurt by things that most of us would react quickly to. For example, Ronald Melzack and Patrick Wall describe a girl who got third-degree burns on her knees by climbing on a hot radiator. There was no signal for her to stop. Researchers are trying to reproduce this condition by genetically altering mice so that they can study the genetic contributions to pain perception.