Armadillos are tough little suckers, but not bulletproof. If you use a shotgun for 'dillo control, you will want to use larger shot, as a previous post suggested. You want to kill them on the spot if you shoot them, if only wounded they have a tendency to leap straight up several feet in the air and take off like turpentined cats- right under some inacessible place, where they then die. And stink. Hoo, do they stink. So use #4s if you have 'em, or bigger. A good accurate scoped .22 is a good idea too, if you have one, again as was suggested previously. I've shot through them end to end with a .22 shooting solids, good hits with a .22 will definitely do the job.
I learned that the hard way though, when they first showed up at home in Alabama several years back my dad complained that he couldn't kill one. Shot 'em with his shotgun, he said, shot 'em with a .22 and just couldn't kill 'em. So next time I came home from NC I brought along a little scoped .223 with 55 grain softpoints. Ol' dillo showed up the first night I was home, mom got all excited cause it was digging up her flowers. So I sallied forth and shot it for her.
Well, I can tell you a .223 is too much gun for dillos. That sucker positively EXPLODED. There were dillo bits scattered all over a 10 foot radius, and I had to scoop up all the dillo parts with a shovel and bury them. And blood? Dillos have more blood per ounce of body weight than any other critter on the planet. Dillo blood stinks too, by the way.
When you blast one, scoop the carcass up with a shovel and bury it promptly, they are nasty critters.