When to plant my blackberry bush?

I live in North west Arizona and recently was allowed to dig up a small blackberry bush that was dying from lack of water. I brought it home in a small pot and watered it for a couple of weeks thinking that it was a goner. Last week we noticed new growth and this week it is doing great. My dh thinks that I should plant it in my garden now to keep it alive, but I didn't know if the cold weather that is on its way would do it in. (It is early November.) Can anybody tell me if I should transplant it now, or wait until spring?