A few ideas:
1. The ring possesses him when he tries to cut off the hand and stops him. He essentially blacks out for a moment, only to realize that he's put his weapon away and his hand is still attached. Every time he attempts to cut it off, the same thing happens.
2. He successfully cuts off his hand, but there's suddenly a bright glowing light and his hand reappears...with the ring right back where it was. The hand he cut off simply fades away as the new hand (or is it the old one...) manifests. Yes, this does mean that the character does have some immunity to losing the limb as long as he has the cursed ring, but he's also cursed, so it's a decent tradeoff for game and narrative purposes.
3. The rat bastard DM option. He successfully cuts his hand off. While looking at his hand on the ground, the ring disapears from the severed hand, and reapears on a finger of his other hand. He still has the ring, and he's lost a hand. If he gets someone to cut off the other hand, it reapears on a toe, ankle, wrist, ear (earing), neck (collar), head (coronet), etc. No matter what he cuts off, it keeps reapearing somewhere else on his body. Even if he somehow kills himself, it stays attached. So even if he is ressurected, the ring is still on him. If he makes a whole new character, the ring somehow finds him also...