Add all ingredients to a medium sized pot and bring to a simmer. Place lid on the pot and let simmer for about 10 minutes. (It’s important to keep the lid on or too much of the water will dissolve.)
Remove from heat and allow to cool for 10 minutes. Strain the simple syrup through a sieve to remove the mint leaves and lime wedges. Chill in fridge until cold. Store in an airtight container, in the fridge, for up to 7 days.
For a Single Mojito:
Place mint leaves and 1 lime wedge into a sturdy glass. Use a muddler to crush the mint and lime to release the mint oils and lime juice.
Add 2 more lime wedges and the simple syrup, and muddle again to release the lime juice. Do not strain the mixture.
Fill the glass almost to the top with ice. Pour the rum over the ice, and then fill the glass with club soda. Stir, taste, and add more simple syrup if desired. Garnish with the remaining lime wedge and a sugar cane stick, if desired
For a Pitcher of Mojitos:
Add the mint leaves, rum, lime juice and mojito simple syrup to a large pitcher. Muddle the mint to release its flavors.
Fill the pitcher most of the way full with ice. Top with club soda. Serve in tall glasses filled with ice, then garnish with extra mint, lime wedges and a sugar cane stick, if desired.