To make the simple syrup: In a heavy medium saucepan, combine the water and sugar and bring to a boil over high heat. Reduce heat to medium-low, stir, and simmer until the sugar dissolves, about 5 minutes. Remove from the heat and cool slightly.

In a 2-quart saucepan, combine the coconut milk, water, lemongrass and ginger. Bring to a boil. Reduce heat to medium-low and simmer for 10 minutes, stirring occasionally. Remove from the heat. Add the simple syrup and stir well. Strain through a fine mesh strainer into a clean container.

Cool in the refrigerator for at least 2 hours. Pour into an ice cream machine and process according to the manufacturer’s instructions. Transfer to an airtight container and freeze until ready to serve.