searched: Wikipedia articles on "confession", "absolution", "contrition", and "penance"
You're a Catholic priest (in English-speaking North America), at least halfway through your career.
A elderly parishioner you've known for years is dying of cancer and has at most a week to live.
They've come to you for Extreme Unction.
In the course of their confession they admit to a (previously unconfessed) grave sin. More accurately, a series of grave sins which spanned decades but which they ceased committing some years back. These acts have serious, potentially even life-threatening, consequences.
Exploring the matter, you come to the conclusion they're not really sorry but are trying to find something which complies with the (minimal) letter of the law without satisifying the spirit.
What are you required, forbidden, and most likely to do?
EDIT: thanks, everyone - I have enough to gnaw on.