Answer: Well established ivy (Hedera helix) can be difficult to eradicate. You can smother it over time using newspaper or cardboard topped with a thick layer of mulch, or you can kill it with glyphosate. This is a nonselective herbicide so you need to apply it carefully; you should also be aware that it can take some time for the chemical to translocate to the roots so be patient and be very sure to follow the label instructions.
You might also want to consider the fact that often ivy is planted as a ground cover in areas where grass is difficult to grow (eg. too shady, poor soil, too steep to mow), so be very sure you really want to get rid of it.
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