Make the rosemary syrup by heating the water, sugar and chopped rosemary leaves in a small saucepan, stirring occasionally, until the mixture is hot and sugar is dissolved. Remove from heat and let the syrup cool completely. Once cool, strain the rosemary syrup into a jar, and refrigerate until ready to use.

Chill a stemmed cocktail glass in the freezer.

Measure the gin, lime juice and rosemary syrup into a cocktail shaker. Fill the shaker halfway with ice, cover, and shake the gimlet mixture about twenty seconds, until very cold. Pour into a chilled cocktail glass. Garnish with a sprig of rosemary or a slice of fresh lime.