The curse cannot die, because the Mariner must retell his story in an attempt to teach others about the dangers of impulsive sinning. When the Mariner sinned, it was on impulse and for his benefit--his motives were purely selfish. Thus, his chance for salvation was destroyed, dooming him to an eternal living death. His soul can never be free from its sin. However, each time he retells his story to an attentive listener, he comes one step closer to the atonement needed for true salvation. The ultimate question Coleridge leaves us is whether that atonement can ever truly be performed.
The curse is a constant reminder for the mariner that he a)caused the deaths of his crew and that he b)committed a horrible act by disrespecting nature and the albatross that is a part of nature by killing it. He must tell his story so that he never forgets the severity of what he did.
The very nature of the curse is that the Mariner must retell his story time and again so that others faced with a situation like his, will choose differently. I do believe that the Mariner gets some personal redemption each time he tells his story if the person who he is telling stays and listens. By the end of the story, the curse is more of a mission.