Heat water up to almost boiling in a small pot over medium heat on the stove.
When water is hot, add sugar and stir to dissolve.
Store homemade simple syrup in the refrigerator.
Muddle the mint in the bottom of the mixing glass, then squeeze the juice of the lime into a mixing glass. Put the squeezed lime parts into the drinking or serving glass. In the measuring glass add the rum, simple syrup, and club soda.
Pour the mix into the mixing glass and stir all the ingredients together. Next, pour the contents of the mixing glass into the drinking glass.
Top off the drinking glass with ice, and you may garnish with a sprig or mint or two. Before placing the mint into the glass, slap it between your hands, it will release the oils of the mint.