I'm going to show how nerdy I am.
Wolverine has been killed and resurrected by Gorgon, and Gorgon just beat him the old fashioned way. To death. Gorgon was beyond normal superhuman, though. According to Sabretooth (who has similar healing), drowning would kill them. Any sort of suffocation should work. Also, when surgery needs to be done on him, they can insert an extremely high radiation capsule to use up his body's healing. Thus, if that was inserted, then he could be killed much easier. Of course, how well he can heal is different in every story line, I'm just talking normal. Xavier could probably kill him instantly; I don't know if Wolverine can heal from mental blocks. Rogues power could kill him; utilize her. There was a girl that had a healing power similar to Wolverine's, and Sabretooth just killed her the normal way. I think that in one show, Magneto's daughter was able to nullify mutant powers. Hurling him into a volcano might work, but he'd have to either be unconscious or pass out from the heat, otherwise there's a chance he could crawl out. And it'd have to be an active volcano.
Of course, there's ways to straight out cheat, like time traveling to when he was a child before his powers were active. I don't know if "Death" exists in the Marvel universe. Temperatures high enough to melt adamantium should work. Cold temperatures might...but there's know way to know if the healing factor would keep him alive after he was defrosted (as we don't fully understand the nature of his power.
A couple of points: To the person who claims it takes a long time for their brains to recover: sometimes this is true, especially in older comics, but in MOST newer comics, the regeneration is much faster, unless something special is done to the brain. Maybe it's possible to hit the part that is responsible for healing?
Someone said that adamantium can pierce adamantium. This might be true, but it's hard enough that you'd probably need super strength to do it. So..I wouldn't try to physically stab him using his own claws yourself. Also, when he has an adamantium skeleton, he has minor super strength, so normal humans cannot overpower him (unless it's convenient for the plot).