At the beginning of Curse of the Black Pearl, the Black Pearl attacks Port Royal to find the last Aztec coin. During the attack, some of the cursed pirates actually die, like the one which Will Turner kills by throwing the axe at him.
If they are cursed, how were they able to die?
No, he did not die.
The pirate in question is the one who throws some sort of grenades and can be seen throughout the movie.
At following time he is shot in back by axe by Will Turner. He lays there unconscious or knocked out. But not dead.
And here, you can see that he's still alive. Will is also amazed at how he is still alive. Notice his expressions.
