Answer: As lovely as it is, Lily of the Valley can become invasive and can be very difficult to eradicate. Digging it out, roots and all is the best control option, but you might also try covering the plants so no sunlight can reach them and they cannot sprout. Weed barrier and mulch come to mind - early in the spring, just as the plants begin to sprout. If they're already up, cut them down at ground level, then cover with weed barrier or even black plastic and cover that with an attractive mulch. Gardeners who have done this report success after a year.
Aside from the above, Round Up will kill the plants and kill the roots, but it will also kill any other plant it comes in contact with. If you keep your pets out of the treated area for a day or two, they should be fine.
Best wishes with your garden!
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