In a blender or food processor pur�e 1 cup of the raspberries with 1 cup of the water and force the pur�e through a fine sieve set over a pitcher, pressing hard on the solids. Add the remaining 1 cup raspberries, the remaining 2 1/2 cups water, the sugar and the lime juice; stir the mixture until the sugar is dissolved.
Divide the limeade among tall glasses filled with ice cubes and garnish each drink with some of the mint sprigs.