Don't forget the hanging execution of "Black Jack" Ketchum in 1901, where the distance he had to drop was miscalculated, and it ended up straight decapitating the guy. But he was an asshole, and he died instantly, so fuck him.
There have also been botched lethal injections before, and it's not always a pleasant way to die, in any case. Different folks have different body chemistries that react in different ways from time to time.
Probably the least painful involves the neck, such as hanging or guillotine, but the former is usually (as Nuri and Tom Brokaw pointed out) viewed as wild west revisited, and the latter is unacceptable in our antiseptic world. The successful snapping of the spinal cord sends the brain into immediate shock and kills it.
However, I'm of the opinion that capital punishment should reflect the crime. You shoot somebody in a robbery? Two in the back of the head. Strangle somebody in the middle of a rape? Choke with a baseball bat up your ass sideways. Crucifixion should be brought out of mothballs for the likes of such irredeemable fuckers like Charles Manson.
But when done right, hanging's as good a way to go as any.
Also, regarding IDE's mention of the hangman's boner,
Wikipedia weighs in:
There is a popular myth about sexual stimulation of hanged men, due to the apparent erection some exhibited. The effect is attributed to gravity causing the blood to settle in the legs and lower torso, thereby engorging the penis. (This myth fuels the auto-erotic asphyxiation, a practice that might lead to an accidental death.)
That means the whole premise behind "Rising Sun" was total bullshit!