Wash ginger root but do not peel. Chop roughly and process in food processor until reduced to a textured paste. Put ginger in a large non reactive container and pour boiling water over it. Stir in the lime juice, zest and cream of tartar. Cover and stir occasionally as mixture cools. When lukewarm, mix the yeast with the warm water, pressing out lumps, and add to container. Cover and let stand for 6 hours. Stir the sugar into the ginger beer until completely dissolved and immediately pour into sterilized bottles, making sure the stoppers are well sealed and airtight. The ginger beer will keep for 3 to 4 days in a cool place. To keep longer, add rum.

To serve, strain and pour into glasses (over shaved ice if desired), with a sprig of mint and thin slice of lime.