Push the raspberries and preserves through a food mill, then strain through a fine-mesh strainer set over a stainless steel bowl to eliminate any remaining seeds. Or puree the berries with the preserves in a food processor, then strain through a fine-mesh strainer into a bowl.
Add the water and mix well. (You should have about 2 cups.)
Place the bowl in the freezer and freeze, stirring every hour or so, until it is half frozen and velvety.
Meanwhile, pour the lime juice into one small saucer and put the sugar in another. Dip the rims of four stemmed glasses (preferably tulip champagne glasses) into the lime juice and then into the sugar, to create a border.
Place the glasses in the freezer or refrigerator until serving time.
At serving time, divide the raspberry velvet among the prepared glasses and decorate each with a sprig of mint.
