TubeNet

Having always played cushion-rim mouthpieces on both tuba and euph, I decided last year to try a Conn/UMI Helleberg with a view to increasing my options. I bought a brand new one and have learned to like it very much, but the wall of the cup is so thin that it perceptibly vibrates, which I would think is a parasitic loss of energy that should instead be going into the air column. My question is do the older Conn Hellebergs have a thicker wall at the cup?

No. They are so thin that they can be dented. They also have less metal around the throat. But they do have a wider rim. I don't care for that, but some folks do.

Sounds like this may be just what the Dr. ordered:
http://www.angus1.com/ssh/index.html

Or wrap a piece of golfer's lead tape around the bottom of the bowl. Cheap. Does the job. I have it on my Kelly for outdoors. It works.