Can I take a cutting of this rose and grow another. What is the best way to do this?
Yes you can. Joseph's Coat's patent has expired and can legally be propagated. For years I used the method, described in the following link, with great success.
Here is a link that might be useful: Rooting rose cuttings
I personally have tried several methods of rooting rose cuttings, but the best way I have tried is our Great grandparents method by using canning jars.
What I do is usually in the fall ( I don't like to try in the spring because it gets too hot quickly here and the rose will suffocate in the jar ) is I take a small cutting that's no wider than a pencil and I do use a rooting powder on the bottom, stick it in the ground and put a canning jar over it. I usually put a few rock around the jar so the wind will not blow it over, and by spring, ta-da a new rose has rooted! I just would not plan on moving it until the next spring though since it's roots are still very tiny and easy to break if you dig it up.
Whatever method you use, I would recommend that you try it with at least 5 cuttings - I do that and usually at least 3 survive. If you get too many, you can pot them up and give them away!