DID YOU KNOW?! Some babies are born and are still in their amniotic sac.
Here's another thing that popular culture has incorrectly taught us - labor starts when your water breaks. Wrong!
You could be in labor for hours before your water (amniotic sac) breaks. Your sac could break and you could not have contractions for a long while. Your doctor may break the sac to speed labor up (popular procedure, called AROM: Artificial Rupture of Membranes).
Or, if your waters are never artificially ruptured or spontaneously ruptured, your baby may actually be born in the caul, as it is called when the baby's head is emerging still inside the amniotic sac. The amniotic sac cushions the baby as it works its way down through the pelvis during contractions. I've seen it in a couple birth videos. The medical practitioner will usually go ahead and rupture the membranes once the baby's head is out so that the baby can breathe.
Being born in the caul is not a bad thing at all, but it is very rare. In fact, a veiled infant it is considered to be lucky or to have some sort of spiritual significance. It does assure that the baby could not possibly have become infected. Midwives are more likely than OB's to allow a baby to be born in the caul.
The wikipedia page on "caul" lists a lot of myths and legends associated with being a "caulbearer."
You can view images of a baby being born in the caul at Birthing Way.