Answer: Ivy can be very difficult to remove. First you would cut the stems at ground level and literally pull it down off the building. The holdfasts that keep it attached to the wall will remain on the wall when you pull the vine down and are impossible to remove; they will leave marks so it is something to consider before you pull it all off. The vines may also bring down loose mortar if there is any. Sometimes, too, ivy is actually protecting the wall surface from the elements; it also provides shade in summer and helps keep the building cool.
Once you tear it down, it will regrow from the roots unless you work at killing the roots. You can try to dig them out by hand but the roots are usually deep and the plants will regrow from little bits of root you miss. You can try to smother them by layer damp cardboard topped with mulch over the root area for a year or two. Clip off and recover any sprouts that manage to come up.
Or, you can spray it with an herbicide containing glyphosate. It usually takes several treatments. With the evergreen English ivy, the herbicide is more effective on tender new growth. The process would be to cut it back, water and fertilize to encourage new growth and let it regrow several feet of new tender foliage, then spray. The spray is absorbed by the leaves and then carried down through the circulatory system to the roots where it kills them. Be sure to read and follow all of the label directions or it will not work. With the deciduous Boston ivy or Virginia creeper, the better time of year to spray is in August to September when the plant is naturally translocating nutrients to the root system in preparation for winter. All in all, it is a big job. Good luck!
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