A vampire's immortality is actually nothing more than a scientific hypothesis. From the moment of their birth, so long as there is blood within their veins, a vampire can heal almost any physical injury in under an hour. The less severe the injury, the quicker the recovery time. The catch to this is there cannot be any obstructions. So if a vampire was to break a bone, the bone would have be aligned and not obstructed by any muscle, tissue, flesh, or other objects that are not supposed to be there. Another great example would be if a vampire was staked. Even if the stake penetrated the heart of a living vampire, if the stake is pulled out before their heart stops beating and their is blood within their veins, they can completely recover within the hour without having to turn into an undead.
Then you have their ability to become an undead. This is when we start getting into the biological makeup of the vampire, as even in the magical community they are very unique. The vampire is the only physical being, human and magical alike, capable of living without the need for a heartbeat. In fact, their heartbeat is what makes the difference between a living and an undead vampire. All vampires are born with a heartbeat, and eventually, assuming their heart stays whole and healthy, they will loose that heartbeat, thereby become a significantly more powerful undead. Once they are an undead, they are no longer susceptible to physical ailments like disease or famine. So the only way they can die is if their heart is irreversibly damaged. This is where the hypothesis of their immortality comes in, as it is believed that under the right circumstances, an undead vampire can live forever. However, this has never been proven.