Removing broken screws from plastic chute

[joesquid]

I've got two broken mounting screws broken flush in the bottle chute of a Vendo 144E.  Of course, the chute is some form of plastic so attempting to drill a hole and bang in an easy out would destroy the chute.  Does anyone have any particularly enlightening ideas?  Are there any drill bits hard enough to drill through the center of a screw?
Curious,

[johnieG]

General assembly screws are rarely anything more than a mild steel (unlike a case hardened bolt for example) a titanium coated drill bit & a steady hand should do the trick, (provideing that your wife doesn't crush it during the delivery!  )  congradulations on No.3 ! are you going to nick-name the little one "Sprite" (bad joke)

[Jim]

Is the door completely apart?
If so, maybe you can get to the screw from behind and start to loosen it enough to finish the removal from the front with vise-grips or equivalent tool...
Otherwise, you'll be forced to follow Johnie's suggestion...