Remove all unpopped kernels from popped corn. Place popcorn in a greased 17x12x2-inch baking pan. Keep popcorn warm in a 300 degree F oven while making syrup mixture.
Butter the sides of a heavy 2-quart saucepan. In the saucepan combine sugar, water, molasses, and salt. Cook over medium-high heat until boiling, stirring constantly. Reduce heat to medium. Clip a candy thermometer to the side of the pan.
Continue cooking over medium heat, stirring occasionally, until thermometer registers 250 degree F (hard-ball stage), watching carefully to prevent the syrup from boiling over.
Pour syrup mixture over popcorn. Add sunflower nuts and stir gently to coat popcorn. Cool mixture until easy to handle. Use buttered hands to shape mixture into sixteen 3-1/2-inch logs. Cool logs completely.
Wrap each log in clear plastic or colored cellophane wrap. Tie ends closed with ribbon. Store at room temperature up to 3 days. Makes 16 servings.