Hi Dave, let me insert answers below:
I don't know how effective this is going to be. The plant is quite
extensive. I've started by cutting the errant stem back to the main stem
that it came off of (it was about 8 feet from the junction), but I think
I'm going to leave that stem temporarily and see what happens.
We don't know how fast the viroids move through the stem - especially
with colder temperatures. IF the viroid movement is faster and has
gotten into the roots, you'll see hideous reddish purple growth coming
from near where you cut it off: either from dormant eyes or axilary
buds, but much greater response to cut back than you would expect from
a normally healthy rose. Remember RRD is a disease of excessive
growth.

I know
that this is a systemic thing and if you think that I'm stupid for doing
this, please say so.
I share your wanting to keep these roses alive, and the speed at which
it spreads within a rose is unknown. I've saved two and lost ..four...
in the past three years and I've tried to save them at this time of
year. Figuring that sap is slower and the mites much less active, we
will try to save them and maybe in a few years be able to predict
sucess better. Do use a systemic miticide to kill any mites that might
still be there.

It just seems like we're going into the dormant
period now and if I'm going to take such drastic action, it should
probably be in January or February. It's going to be really hard to
isolate this as all of the stems are really close at the
ground level. The plant iself has a spread of about 20 feet on a 4 foot
high chain-link fence but they all come from a fan shaped base growth
area.

Here's a pic of the entire plant from early April:

http://www.pbase.com/image/23294494

And here is the plant in its full glory:

http://www.pbase.com/image/23294847

In the first photo, the circle is the area where the ground growth is,
the straight line is the demarcation between it and a The Fairy (Cl),
which of course I'm worried about now, and the triangle is the
approximate location of the affected growth (which has now been cut off
back to the main cane that the cane that it grew off of grew
from).

Here's a pic in March, which shows the structure of the plant a little
better. I'm having trouble envisioning how I'm going to go down below
ground level without taking the whole plant up.

http://www.pbase.com/image/23294779