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Cut the watermelon into quarters and remove the pips using a teaspoon. If you’re serving the same day or next day, keep six wedges or slices back for decorating and chill. Cut the flesh into 3cm cubes (you should have about 1kg fruit, with the rind discarded) and sit in a baking parchment-lined container where the cubes fit snugly. Pour the prosecco over the fruit, cover and freeze for 4 hrs, or up to 1 week.
Put the mint, lemon and sugar in a small pan with 50ml water, and bring to a gentle simmer for 3-5 mins until the sugar dissolves and turns syrupy. Cover and leave to cool completely, then strain.
Put the frozen prosecco-soaked watermelon cubes in a food processor with the sugar syrup. Blend well, stopping and pushing the mixture towards the blades with a spatula every few seconds, if you need to, until it’s a slushy consistency. Add a splash more prosecco to make a smooth slush. Spoon or pour into six chilled glasses. Garnish with the chilled watermelon wedges and little mint sprigs.