In a medium saucepan, combine sugar, ginger, and 1 quart (945ml) cold water. Bring to a boil over high heat and cook, stirring, until sugar is dissolved. Remove from heat, cover, and let stand 1 hour to steep.
Strain syrup through fine-mesh strainer and funnel into 2-liter soda bottle. Add lime juice, then fill with cold water to within 2 inches of top. Cap and cool in refrigerator or ice bath until about 65°F (18°C).
Add champagne yeast. Cap bottle and let stand at room temperature for about 2 days, checking bottle pressure intermittently by squeezing it or releasing the cap slightly and briefly.
When ginger beer has achieved desired carbonation level, transfer to refrigerator. To serve, strain into glass.
