Take the cuttings either while it's blooming or shortly afterwards.

The hard part isn't rooting the cuttings, but getting them through their first winter.

Thanks for the clarification! I'm not overly worried about the winter actually. The bush has survived -20 temps with no protection. I'll mulch the babies and put them in the garage and dump some snow on them a few times and we'll see what happens. Thanks again!
Karen

This is a picture of my unknown pink rambler just blooms once also.

I covered some canes up during the fall to distribute more of this wildness to other parts of yard.I tried some after flowering last year.no luck.but the cuts I made removing a cane a chopping it into 1 ft pieces....did good.The ones with the oldest growth we're what took.All I did was shove in a container of potting mix& it sat there all winter on potting table.
I know the rose has about engulfed the fence its on& is in its 3rd year.I've never trimmed it.It will bloom in a few weeks(Being south TX)
Sorry I only took this one lone picture last year.But you'll see its glory soon.