Rinse and stem the black mulberries, then gently pat them dry.
Combine the water and granulated sugar in a small saucepan over medium heat. Stir until dissolved, bring to a brief simmer for 1 minute, then remove from heat and cool completely in the refrigerator.
Blend the black mulberries with the lemon juice until very smooth, 30–60 seconds.
Pass the puree through a fine-mesh sieve into a bowl, pressing to remove most of the seeds; discard the seeds.
Stir the cold syrup into the mulberry puree until evenly combined, then chill the mixture until very cold, about 1 hour.
Pour into a shallow metal pan (such as a 9×13-inch baking pan) and freeze until the edges begin to firm, 30–45 minutes.
Scrape and stir the mixture with a fork, breaking up the ice into fine flakes. Return to the freezer.
Continue scraping every 30 minutes until the granita is evenly icy, fluffy, and spoonable, 2.5–3 hours more. The flakes should hold their shape but melt cleanly on the tongue.
Fluff with a fork and serve immediately in chilled glasses. If firmer, let stand at room temperature for 5 minutes and rescrape before serving.