"It is not the voltage that kills you, it is the current" is an electricity safety tip I've heard many times.

For example, here is an experienced electrician making the claim:

The real danger with electrical shock is amperage, not voltage. Although people have been killed by as little as 120-volt AC and low voltage DC, the silent killer is the amount of amperage in milliampers that flow through a person's body. Any electrical device is capable of dealing a deadly blow, depending on the amount of current flowing through it.

and a claim by New Jersey State Council of Electrical Contractors Associations:

It's The Current That Kills

Offhand it would seem that a shock of 10,000 volts would be more deadly than 100 volts. But this is not so! Individuals have been electrocuted by appliances using ordinary house currents of 110 volts and by electrical apparatus in industry using as little as 42 volts direct current. The real measure of shock's intensity lies in the amount of current (amperes) forced though the body, and not the voltage. Any electrical device used on a house wiring circuit can, under certain conditions, transmit a fatal current.

and a typical forum post:

No its current thats dangerous. You can generate 20,000 volts of static electricity walking across a carpet and get zapped a littel but its not going to hurt you. Take a low voltage, High current, and good conductor path and you're dead meat.

So, which is it? Is it the current that kills you or the voltage?