There are few characters in comics as vile, depraved and demented as the Joker, and despite Batman's best efforts, the criminal antics of the Clown Price of Crime seem impossible to stop. No matter how many laws he breaks or how many people he's killed or seriously injured, the Joker will invariably (perhaps inevitably) be sent to Arkham Asylum or Blackgate Prison, from which he'll promptly escape. The Joker seemingly never dies, even in a country with the death penalty... but there's a reason for that.
The Joker has "died" plenty of times in the world of DC comics, but he first shuffled off the mortal coil in Detective Comics #64; death by electric chair didn't stick. From there, a vast majority of the Joker's subsequent "deaths" are simply fake-outs and parlor tricks. But the actual reason why the Joker can't die in Gotham City is simple: in the DC Universe, Gotham has no death penalty.
In The Batman's Grave #2, Batman and Jim Gordon meet on a rooftop in the rain - par for the course for a Batman story. The Commissioner and the Dark Knight talk about Eduardo Flamingo, a criminal with cannibalistic tendencies. Through dialogue spoken by Gordon, the reader realizes that Gordon is differentiating between Gotham and other western cities in California. "But San Diego, Los Angeles and Sacramento all want him too, and California has the death penalty."
One can thus infer that Gotham City does not have the death penalty. In the DC Universe, Gotham is canonically located in New Jersey, which abolished the death penalty in 2007. Even before that, the last execution in New Jersey took place in 1963. Given the Joker's extremely high body count and numerous other crimes, it's likely that a jury in a state with the death penalty - for instance Oklahoma, which has one of the highest rates of executions in the United States - would find him guilty and sentence him to death.
Batman, regardless of the state out of which he operates, would no doubt adhere to his so-called One Rule, and wouldn't kill the Joker. But once turned over to the police, the fate of the Joker would be entirely out of Batman's hands, and his death would be all but certain. The Joker's life might be safe in Gotham, but were he to be convicted elsewhere, Batman's mortal enemy would be considerably more mortal.