Question
Today, I cut a snippet (12 inches) of a rose tree I planted 5 years ago in memory of my mother. As we've sold the house, the current owners allowed us this "snippet".
Can I quickly graft this branch to another tree, bush, or plant to keep it alive?  Thanks much in advance; I know virtually nothing of this subject, but it suddenly got painfully important!

Answer
To grow the cutting well it should be taken from the end of a cane that has flowered in the early spring or early fall. Taking a cutting now makes it tricky to start. You should also leave the top two leaves on the end of he cutting. However if you can't then just follow the directions for the next two ways of making a cutting grow. The current owners of the house should realize that your mothers rose will have to be pruned in the spring and perhaps at that time you could get some more cuttings off the bush.
Pop-Bottle Method
Construct "greenhouses" for the cuttings using empty two liter soft drink bottles small pots or cups. use a knife to cut off the bottom of the bottle where the solid-colored section is joined to the clear section. Discard the bottom. Keep the bottle cap. Use pots which fit into the bottle bottom. The top of the bottle should just fit snugly inside the pot rim. Place the top from the soda bottle over the cutting and press it gently into the soil. Make sure that there are no gaps through which evaporation could occur. Put the cap on the bottle, but do not tighten. After a few hours check the bottles to make sure there is some moisture condensed on the inside of each. If not, your soil may not have been moist enough and a small amount of additional water should be added. Place the bottles in a bright light but not in direct sunlight as below. Because you are doing it in the winter where there will not be enough light, just place the cutting as described above but make the soil only damp and then place it in a cold unheated area. Continue the instructions when the weather warms up in the spring.
You can make two 6 inch cuttings out of your rose cane for the following method.

Baggie Method

Use large zip lock clear plastic food storage bags to create the greenhouse atmosphere. Fold the top of the bag down several inches in order to keep the zipper clean while filling the bag. Place some moist potting soil into the bag. Firm both soil and bag into a ball shape with a flattened bottom. Use a stick to poke a 2" deep hole in the soil not the plastic. Treat the cuttings with rooting hormones. Insert the cutting. and firm the soil tightly around it. Unfold the top of the bag and close most of the zipper, leaving the last inch open. Inflate the bag to expand it then close the remaining length. Finally, place the bag in indirect light as below. Place the cutting pots in a place where heat does not heat the cuttings. The cuttings should have bright light to allow photosynthesis. In the spring many varieties will develop roots in ten days to two weeks. Some varieties can be rooted in three to four weeks. In the winter roots may form in four to eight weeks. To determine if a cutting is rooted you can tug lightly on it; if it resists being pulled out of the pot it is likely rooted.
After all those instructions, if you are living in an area that gets a warm winter, you can go ahead and following the instructions as written. Because you will probably have enough warmth and light to start them.
Here is a site that tells you about taking cuttings. A bit wordy but you will get some more information off it.
http://www.rdrop.com/~paul/hulse.html