Combine the sugar with 1 cup of the water in a small saucepan and place over medium heat. Bring to a simmer then kill the heat. stir occasionally, until the sugar has dissolved. Let it sit until it cools to around 170℉. Transfer the syrup to the carafe of a blender.
Meanwhile, combine the ice and 2 more cups of water in a large bowl and have it standing by near the cooktop.
In the same saucepan or an electric kettle, bring several cups of water to a boil. Tie the mint into a bunch with cotton butcher’s twine and plunge into the water for about 15 seconds then immediately plunge the mint into the ice water to stop the cooking.
Once thoroughly cooled, pat the mint dry, pluck as many leaves off as possible, and add them to the cooled syrup. Blend on medium speed until the mint is reduced to very fine pieces. Strain through a fine sieve set over a 2-cup liquid measuring cup then transfer to a plastic squeeze bottle and refrigerate for up to 2 months...or freeze forever.
To use the syrup in say, a mint julep, combine 1/2 fluid ounce of it with 3 fluid ounces bourbon in a rocks glass filled with crushed ice. Stir, then garnish with a fresh sprig of mint.