Some 800 Americans make their living from bounty-hunting - tracking down and capturing absconded defendants. A good bounty-hunter will make $60,000 (£40,000) a year.
It is a privatised style of law enforcement that sits well with the Republicans' notion of getting away from big government and returning to the old-fashioned American virtues of individualism and self-reliance.
Mr Gingrich, the Republican Speaker of the House of Representatives, has met with the bail industry's representatives and promised to defend them against efforts by the Clinton administration to shift the bail business into federal hands.
The venue for the first ever bounty hunters convention was the Bella Union, an opera house-cum-saloon founded in 1881, the year of the shoot- out at the OK Corral. The defining drama in the history of Tombstone, heart of the Old West, took place around the corner from the Bella Union, within easy range of Wyatt Earp's Smith and Wesson 45.
But the spot where Wyatt, his brothers Virgil and Morgan and John "Doc" Holliday faced down and slew three members of the infamous Clanton gang was not, in fact, the OK Corral, as Bob Burton, the bounty hunters' president and master of ceremonies, pointed out. "It really took place in an empty lot next door but `the shoot-out at the empty lot' just doesn't have it, does it guys?"
The guys cackled. On looks alone the Clanton boys would have taken all of them to their black hearts. There was Ray Hawkins from California - black hat, black jeans, black waistcoat, black boots, black beard, black pony-tail, crushed nose, moll in tow with stiletto heels; Ken Bishop, world champion chicken-plucker from Colorado, red braces, red face, Kentucky- fried belly.
Then there was Hank Hustus from Tucson, six foot four, built like a house, melancholy Earp brothers' moustache; Mel Barth from Virginia, beefy Vietnam vet who flew planes over Nicaragua for the Contras, was arrested and testified at the Iran-Contra hearings; and "Bad" Bob Burton himself, a burly old rogue in a bushy white beard who, among a host of dubious accomplishments, spent six months in the late Seventies teaching the South African and Rhodesian security forces the techniques of "improvised hostile interrogation".
Tough nuts all, to a man they fell silent when "One-eyed" Luke took the stage. Luke was the real thing: save for the patch over his right eye, the spitting image of Virgil Earp.
Gaunt, grey, straight-backed, Luke touched the peak of his black Stetson hat and began, gravel-voiced, to recite his poem: "Predators stalk the night/Innocent nervously feed upon the land/The game is not a game/When man is hunting man...Instinct lengthens life/This is nature's plan/Instinct is the keenest/When man is hunting man."
The dream faded when the first speaker informed the assembled hunters, 70 in total, that they were on the cutting edge of Newt Gingrich's Republican revolution.
It dissolved altogether when speaker number two elaborated on the paramount importance of knowing the regulations required by USAIR, Delta and Continental to carry firearms on board. "One problem is that some airlines won't let you escort a prisoner cuffed."
What does America's late 20th century bounty hunter do? "This," said Hank Hustus, "is how it works: a judge charges an individual and sets bail; the individual goes to a bail bondsman who'll come up with the bail in exchange for assurances of collateral - like a car, a house - and an undertaking that in the end he will get 10 or 15 per cent of the bail amount in commission. Then the guy doesn't show up in court on time and this is where we enter the picture.
"The bail bondsman, who would rather avoid paying the, say, $10,000 than dealing with the collateral, contacts one of us to track the fugitive down.
"We'll normally get 10 per cent of the bail amount, more if the guy is very dangerous. But no body, no booty. If you don't find your man, you don't make a cent."
Bob Burton, who has carried out more than 3,000 arrests in 23 countries during a career spanning 27 years, is confident that history is on the bounty hunters' side. He believes they provide a valuable service conducted - free of charge to the taxpayers - with a minimum of violence.
Which is not to say that "Bad" Bob dislikes his image. A sign on the entrance to his Tombstone home reads: "If you come through this door you will be killed". He has pictures on his wall celebrating his feats in South Africa and Vietnam, where he served as a marine sergeant. In one photograph he poses with Robert De Niro, whom he coached for his bounty hunter's role in Midnight Run.
"Yet the truth is that this job requires more brains than anything else. My phone is my main weapon. I've shot at a few people. I've punched and kicked and knocked people out.
"But those are the exceptions. We always survey a scene before we pop a guy, I mean catch him. We literally choreograph the arrest to minimise the risk - one reason why we get sued far less times than the police.
"You've got to be smart. Once I arrested a guy by posing as a rabbi. This Jewish guy skipped bail who'd been doing a big money scam out East.
"I went to his mother's house in West Hollywood dressed as a rabbi, praying she wouldn't speak Yiddish to me because I'm a good Catholic boy. She was taken in, she actually called me `rabbi', and through a few tricks I got her son's phone number in Vegas.
"I phoned a friend who got me the address. Next morning I drove there, saw him outside his condo and I arrested him. Easy. In less than 24 hours I made $8,000 on an $80,000 bail."
"Bad" Bob won't make that sort of money in Tombstone, a town of 1,500 people with one of the lowest levels of crime in the United States. No one has been murdered since 1987. Daily re-enactments of the OK Corral showdown for the tourists offer the nearest thing to street violence nowadays.
The bounty hunters went out to watch on Thursday afternoon. "One-eyed" Luke gave the show a miss. He was out on the range, breaking in a mule.