Method

Peel a chunk of the ginger with the tip of a teaspoon (the papery skin scrapes right off) and grate it, using the fine side of your grater. Place the ginger, sugar, and water in a saucepan over medium heat and stir until the sugar is dissolved. Turn off the heat and allow the mixture to steep for an hour.
Strain the mixture (discard the ginger solids) and allow to cool.
You have now made ginger syrup. Place a funnel in the top of the bottle. Sprinkle the yeast in, followed by the syrup, lime/lemon juice and zest, and water.
Put the lid on the bottle and shake the concoction until the yeast is dissolved. Stow it on a shady shelf out of direct sunlight for 2-3 days, or until fizz is achieved. At this point it is ready to drink, and must be stored in the refrigerator to prevent further fermentation. Don't forget about the bottle, or the pressure will build up so much that it may explode!
As with any yeast-powered beverage, the fermentation process continues unless you prevent it from happening. Refrigerating will slow the process down but not stop it completely.
Consume within 1-2 weeks.

CAUTION: be sure to open the bottle every day to release the extra gas, otherwise the bottle might explode!
Note that as the beverage ferments, sediment will settle at the bottom of the bottle. You can strain it out if you wish.

Serve over ice and savor the spicy taste of your very own homemade ginger beer!