1. In a blender container or food processor bowl combine half of the cut-up melon and half of the margarita mix. Cover and blend or process until smooth. Transfer to a medium bowl. Repeat with remaining melon and margarita mix. Stir in mint, lime juice, and, if desired, tequila.
2. Pour melon mixture into a nonmetal freezer container. Cover and freeze for at least 4 hours or up to 24 hours, stirring occasionally. (The consistency should be icy, not smooth.) To serve, scoop into 4 margarita glasses or small wine glasses. (If too hard to scoop or scrape, let stand at room temperature about 15 minutes or until slightly softened.)