In a medium pot combine 1 cup of the water, ginger, peel from the lemon or lime, molasses and sugar. Bring to a boil on medium heat and lower to a simmer. Cover and cook for 10 minutes.
Remove from heat and if desired, allow to steep further. Otherwise cool slightly for 10 minutes and add the remaining 3 cups water. If necessary, cool further so the mixture has no warmth left, otherwise the heat can kill the ginger bug.
Add ginger bug and mix well. Strain the ginger ale through a fine sieve and transfer to sterile bottles. Fill only 2/3 full. Seal tightly and allow to ferment 2-5 days (cooler climates can take longer). Burp bottles occasionally during fermentation to allow gas to release.
Once bubbly, store in refrigerator for up to 1 week. Ginger ale gets more bubbly when it sits out of the fridge for several minutes before serving.