Additional Study Information: This study will evaluate Patent Foramen Ovale (PFO) closure for migraine headache relief using a device called the GORE CARDIOFORM Septal Occluder. A patent foramen ovale (PFO) is a hole in the heart that didn't close the way it should after birth. During fetal development, a small flap-like opening the foramen ovale is normally present in the wall between the right and left upper chambers of the heart (atria). Most patients with a PFO do not have any symptoms. However, the condition may play a role in migraine headaches and it increases the risk of stroke, transient ischemic attack (stroke that lasts for a few minutes) and heart attack. If the hole is not closed, it can sometimes cause migraine headaches in some people. The PFO can be closed by a permanently placed device that is put into your heart through a small hole in your groin. It is not known if closing the PFO will reduce migraine headaches. It is possible that there will be headache relief from this procedure, and it is also possible that there will be headache relief from a medication given earlier on in the study. It is also possible that neither the procedure nor the medication will provide headache relief.