Just got this finished. Mostly luck will get this achievement, except for the guide from XCVii007r1 (which was done very nicely I might add).
I finished with 2 other guys, we had 1 hobbler running around and when we felt we were ready (we had a feeling the Thief was coming out after the last zombie died during this round as the power had been turned on for a level before) we killed the last hobbling zombie to start the Extra Thief Round. Right after killing the hobbling zombie, he dropped the Death Machine. One of our team retrieved the Death Machine and the thief popped up to only be shot down in 1 sec from the Death Machine.
Now onto the non-luck howto guide:
In the below video, Upon hearing the alarms going off (which only happens when the thief will appear), XCVii007r1 in essence is killing himself (while having a Revive) with a cooked grenade. The thief then appears seconds later. While your dead, you have dual laser grenade launcher pistols (very very high damage), and within mere seconds the Thief is downed.
Many thanks goto XCVii007r1 for the below video guide. Enjoy and Happy Hunting...