I'm trying to perfect the technique on this one, but no matter what I do, the second my finger leaves a string, it keeps ringing out; even at lower gain levels.
Is there something special that needs to be done, or is palm-muting the only way to get rid of the ringing?
Your right hand is the key. Listen to the song I posted in bitwise's "talent thread" a while back. It's only a 3-string-arpeggio but it turns into a flurry of notes, cleanly picked with sweep picking, as soon as you learn to use the right hand correctly.