Finely grate the celery rib into a mesh strainer set over a small bowl. Press on the pulp to extract the juice. Pour 3/4 ounce of the celery juice into a shaker and add the gin, simple syrup, lime juice and mint leaves. Add a large handful of ice; close the shaker and shake vigorously. Strain the cocktail into a chilled martini glass and garnish with the mint sprig.
To make simple syrup, simmer 1/2 cup of granulated sugar with 1/2 cup water in a small saucepan over moderate heat, stirring, until the sugar has dissolved. The simple syrup can be refrigerated in a covered container for up to 1 month.