Make a simple syrup:  Combine the sugar and water in a heavy, medium saucepan and bring to a boil over high heat. Reduce the heat to low and simmer until the sugar is completely dissolved, about 5 minutes. Remove from the heat and let cool slightly. Transfer to a container to cool completely in the refrigerator, about 2 hours.

Purée half of the strawberries, simple syrup, and lemon juice in a food processor on high speed until very smooth. (Alternately, use a blender, or place the ingredients in a large bowl and puree with a hand-held immersion blender.) Place a fine mesh strainer over a large bowl. Slowly pour the purée through the strainer, pressing down on the solids with the back of a soup spoon to extract as much liquid as possible. Discard the seeds and pulp remaining in the strainer. Repeat the process with the remaining ingredients.

Process the purée in an ice cream machine according to manufacturer's directions. (This will take 20 to 25 minutes.) Transfer to a plastic container and freeze until firm.

Enjoy.