In a medium saucepan, bring water to a boil. Remove from heat and add fresh ginger. Steep for 30 minutes. Bring water back to a boil and stir in sugar. Simmer for 5 minutes or until sugar has dissolved. Strain whole ginger pieces out of the simple syrup.
Mix ginger simple syrup, ginger beer, lime juice and vodka.
Pour Moscow Mule mix into popsicle molds and freeze for at least 8 hours.