Lay a double piece of cheesecloth on a nonporous work surface (as the berries will stain a wide array of cutting surfaces and clothes, this may be best done outside or over newspaper and wearing an apron or smock).
Place the blackberries on top of the cheesecloth and gather into a bundle like a hobo sack.
Hold the sack of berries over a glass, stainless steel, plastic, or ceramic bowl; twist the top of the sack to squeeze the juice from the berries into the receptacle (this will yield about 1 cup very strong, tart, dark juice).
Refrigerate the juice until needed; discard the purple mash.
In a small saucepan, combine the sugar, 1 cup water, the lime leaf, and the cardamom pod; bring to a boil.
Simmer over low heat for 10 minutes, or until the mixture is reduced to a thin syrup.
Remove the lime leaf and cardamom.
Allow the sugar syrup to cool and then chill it.
In a 1 quart pitcher, combine the blackberry juice, sugar syrup, and lime juice; stir to combine and then refrigerate until cold.
To serve, stir the ginger ale into the pitcher; fill glasses with ice, and pour in the blackberry limeade.