I have a Samsung cell phone, model sch-a670.

No matter what mode i have it in (silent, vibrate, etc) it always
leaves a short audible ring after someone leaves a voicce messege.

How do i turn this ring off?

My service provider is Verizon in Washington DC

I have the same phone and issue. The only way that I can stop the
phone from ringing when I get a message is by having the phone on
silent mode. Do this by holding down the # key until the phone
vibrates. This will override any other sound settings. The phone will
vibrate for calls, text messages, etc.

In the settings there is an alert for voicemail, by default it is
sound, you need to choose silent, or do as Socca suggested...