Chickweed (Stellaria media) is an extremely hardy weed. There are many different varieties of chickweed, with the most common being appropriately named "Common Chickweed". All chickweeds are known to carry plant viruses and house insect pets, and some varieties of this annual plant can survive the winter. In addition, chickweed seeds can germinate in almost any climate and in any type of soil. Most gardeners use a combination of three different types of toxic herbicides to kill this stubborn weed, but you can also kill chickweed without any chemicals or by using an organic herbicide.
Work on a dry, cool day before the chickweed has flowered. Make sure the soil is dry, because wet soil will cause any missed seeds or stem nodes to germinate.
Pull it up by hand. If you only have a small amount of chickweed, the best way to organically kill it is to remove it by hand. Slip on some gardening gloves and carefully place all bits of the plant in a plastic bag, as chickweed stem nodes can actually re-root.
Lay two inches of organic mulch down in the areas where you removed the chickweed by hand. The organic mulch will limit sunlight that may reach missed seeds, and also "weigh down" any seedlings trying to sprout.
Solarize large swaths of chickweed during the hot summer months. Cover the chickweed with plastic mulch (sheet mulch is most effective). Leave the mulch in place until cooler weather settles in. The extreme heat generated under the plastic will kill the chickweed and the seeds as well.
Target chickweed within a lawn or near other plants with an organic herbicide. There are many on the market, but you can also reach into your pantry and use white vinegar. The higher the acidic level in the vinegar, the more effective it will be in killing the plant. Place the vinegar in a spray bottle and spray it directly onto the chickweed. Repeat once per week until the plant dies.