Combine the sugar with 1 cup water in a small saucepan and heat until the sugar dissolves completely. Set aside to cool or refrigerate for up to 2 weeks.
When ready to make the cocktails, rub the cut lime on the rims of 6 glasses and lightly dip the rims in a plate of sugar. Turn the glasses upright and set aside to dry.
Combine the lime juice, 1 cup of the sugar syrup, and the rum in a pitcher. Fill a cocktail shaker ¾ full with ice, pour in enough cocktail mixture to cover the ice, place the lid on the shaker, and shake for a full 30 seconds, until very cold. (Time it; it’s longer than you think!) Fill the prepared glasses ¾ full with ice and strain the daiquiris into the glasses, garnish with the lime slices, and serve ice cold. Repeat until you’ve used all of the cocktail mixture.
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