This answer is based on a rules-as-written interpretation of the rules. There is an open FAQ request on this subject at paizo messageboards, asking if trolls can be killed by death effects, currently at 95+ votes. The only evidence I found that supports the opposite is a flavor text in rise of the runelords:
Although the trolls recovered quickly from the ogre attacks, several of them perished when the ogres hit on the idea of throwing trolls over the edge of the dam once they were beaten unconscious in battle, drowning the trolls before they regenerated back to consciousness.
No, you have to stop the regeneration
Creatures with regeneration cannot die:
A creature with this ability is difficult to kill. Creatures with regeneration heal damage at a fixed rate, as with fast healing, but they cannot die as long as their regeneration is still functioning (although creatures with regeneration still fall unconscious when their hit points are below 0).
But if you can stop the regeneration for a single round, they can be killed:
Certain attack forms, typically fire and acid, cause a creature’s regeneration to stop functioning on the round following the attack. During this round, the creature does not heal any damage and can die normally. The creature’s descriptive text describes the types of damage that cause the regeneration to cease functioning.
Simply preventing the healing from regeneration doesn't work either, as evidenced by the Block Upper Chakras feat, you have to prevent the ability from activating at all.
So, normally, what happens is that we hack trolls to pieces, then someone lights up a torch and stops their regeneration with 1 point of fire damage, which immediately kills the creature since they are by now around -100 hit points if nobody stopped our fighter-type character.
A creature with fast healing, however, could be killed easily using death knell.
This is a tricky question, but in my opinion, yes it should.
The reason for this is under the description of regeneration
Attack forms that don’t deal hit point damage are not healed by regeneration.
Regeneration is meant to work very well against damage attacks. It mentions nothing about other means of killing it, which seems like a very large oversight if it was intended to also work against those. Creatures are supposed to be hard to kill, not impossible.
It does (or should).
My conclusion comes from the Tarrasque regeneration that specifies:
(...) it regenerates even if disintegrated or slain by a death effect. If the tarrasque fails a save against an effect that would kill it instantly, it rises from death 3 rounds later with 1 hit point if no further damage is inflicted upon its remains. (...)
So apparently the mighty Tarrasque get a special power that makes it recover from death effects in 3 rounds. If you consider that regeneration already makes one immune to death by a death effect it is a power-down worded like a power-up.