Cover whole stems lightly with a fine spray of water, wrap loosely in a paper towel, and place in a plastic bag. Store in the vegetable bin of your refrigerator.
You can also trim the stems, place in a glass with an inch of cold water, loosely wrap the top with a damp paper towel, and invert a plastic bag over the top before storing in the refrigerator. Water should be changed every day or two.
Fresh dill sprigs can also be frozen for up to two months. The dill will darken a bit in color. No need to thaw prior to use. Frozen dill will have more flavor than dried, but not as much as fresh.