Crush the allspice berries in a mortar and pestle (or put them in a ziplock bag and crush with a heavy skillet). You want them cracked and coarse, not pulverized into dust.
Pour the crushed berries into a glass jar and add the rum. Seal the jar and shake well.
Store the jar in a cool, dark place for 4 days, shaking it daily.
On day 5, add the cinnamon stick to the jar. Let it steep for another 8 days
Strain the rum through a fine-mesh sieve, then run it through a coffee filter or cheesecloth to remove the fine sediment.
In a saucepan, combine the water and brown sugar. Bring to a boil over medium heat, stirring until the sugar dissolves. Let it boil for 2 minutes, then remove from heat and let it cool completely.
Pour the infused rum into a clean bottle or jar, then add the cooled sugar syrup. Stir or shake to combine. Let the mixture rest for at least 24 hours before using.