You can do it anytime although fall and spring are best. Decide where you want the rose first and get that hole dug first. Keeping the root ball together is best but if it breaks up it is not the end of the world. Try not to cut the main roots off too short. Just keep the bush well watered after you move it.

Keep the root ball as big as you can.
And when you get it into the new hole, WATER.

We have mostly done this WITHOUT cutting the plants back, but I think most recommend
that new growth be cut back.

We found that the plants bounced back after a couple of days of good watering.
Even in hot weather.

Jeri

I cut mine back severely. The root system is as big as the top growth, but it's mostly composed of tiny feeder roots that are left in the soil. The smaller root system left when the rose is removed will only readily support a like sized amount of top growth.
I choose to prune back before I transplant to avoid having to do it a little at a time as portions of the top die.
By removing a large portion of top growth now, as soon as the bush gets established it will grow back and you'll end up with a similar bush to the one you had a lot sooner and with less fustration. As stated by others, water, water, water, deeply and often. When you think you've watered enough, water some more. Just make sure you have drainage. Once in the ground the roots don't want to sit in water.
If the soil falls off the roots when dug up, soak it as a new bare root for 24 hours before replanting.
To move the rose and avoid losing any soil from the root ball, place it on a tarp and drag it to it's new location.
If you can choose a time when rain and high humidity is forcast for an extended priod, your transplant's chance's of survival will be much better.
Don't fertilize until it new growth appears and the bush looks like it's OK. Then use a very dilute mix or an organic product to avoid burning the new feeder roots.

The easiest is late winter before it leafs out. I cut the canes back to ~ 1 1/2 feet in length, just to make it easier to handle. I like the tarp idea for a move in the same yard. If its a longer move, a large plastic pot is the best. A hand truck is a great idea for a heavy rootball. I have moved roses at all times of the year, but I think late winter-early spring is safest.

After you dig a big hole to put the rose in, I recommend filling up the hole with water and let it drain down before putting the rose in the hole, rather than watering only after you put the rose in the hole. You will have a lot more water available to the transplanted rose by doing it this way.

I think I'll wait until after winter and follow the tips you provided. Thanks to everyone that took the time to respond.