Combine 1 cup water and 1 cup sugar in a heavy saucepan.

Bring to a boil over high heat to make simple syrup.

Reduce heat to medium-low, stir, and simmer until the sugar dissolves (about 5 minutes).

Remove from heat and allow to cool slightly.

In a medium saucepan, combine coconut milk, 1 1/2 cups water, chopped lemongrass, and minced ginger.

Bring to a boil.

Reduce heat to low and simmer for 10 minutes, stirring occasionally.

Remove from heat.

Add the simple syrup to the coconut milk mixture and stir well.

Strain the mixture 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.