This isn't precisely canonical, but here's how I'd house-rule it.
The item effectively casts a spell on whoever activated the curse (in this case, used the sword in battle). Break the item and its ability to cast curses ends. Remove Curse on the character and the curse on _him_ ends (although it can be renewed if he's foolish enough to activate the curse again).
Now, that said, I prefer to have curses be a matter of moral retribution more than anything else. Harass the old woman - ie, needlessly and unlawfully encroach on her personal sovereignity - and she has the _right_ to curse you and make it stick. Rob a grave and any curses against graverobbers aren't even going to slow down for spell resistance and saving throws. Anything where the victim didn't earn the curse is just a magical compulsion and should be zappable with dispel magic.
(A sword that belongs in a grave might give a grace period to someone who picks it up without knowing what it is - but as soon as the sword can arrange for the appropriate info to be available, the curse hits if the new user doesn't make an effort to return the blade to its rightful resting place.)
Oh, and, first post on this board, hi everyone.
-Albert