Bash the thick end of the lemongrass with the back of a cleaver (to release the flavour). Put sugar, lemongrass and ginger in a saucepan. Pour in water and bring to boil.
Reduce the heat and simmer for three or four minutes. Stir occasionally until sugar is completely dissolved. Add screwpine leaves and continue to simmer for a further three to five minutes. Remove from the heat and leave aside to cook completely.
Divide rum between four individual glasses and squeeze in the lime. Pour in lemongrass syrup and stir briskly.