Combine chilled sugar syrup, pomegranate juice, and lemon juice. Refrigerate until cold.
Process chilled mixture in an ice cream machine according to manufacturer’s instructions. Process to a soft-serve consistency, then transfer mixture to a storage container. Press a piece of plastic wrap onto surface, seal container and freeze until very firm.
To serve, scoop into chilled martini or champagne glasses, garnish with fresh pomegranate seeds if desired, and serve immediately.