Answer: You have several options. You can continue chopping off the top and hope that the root runs out of energy before you do. You can try to "smother" it by chopping off the top as close to the ground as possible, covering it with several layers of newspaper or cardboard and top that off with a thick layer of mulch. Finally, you can attack it with an herbicide such as glyphosate being sure to follow the label instructions exactly. Since this is a non-selective herbicide, you will have to be very careful not to get any on the rose. (Some gardeners are able to configure a temporary shield of cardboard, for example, during the application process.)
Good luck with your weeds!
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