If you cut the shrub off close to the ground, wrap it in a black plastic
garbage bag. A couple of layers of the black plastic should restrict
light enough to help. However, the shrub will probably put out shoots.
You will have to cut them off, probably once every week or two.
Otherwise they may poke holes in the plastic. You might have to replace
the plastic bag occasionally, but they're cheap.

Depending on the shrub, you may have to keep this up for two or three
years (probably not in the winter, depending on where you are) to
completely discourage it from growing.

If you can dig around the roots a bit, cutting the roots below ground
will speed up the process of killing the shrub. The deeper the better.
However, that makes it more difficult to apply and maintain the plastic bag.