Place the lime pieces, pulp side up, in the bottom of an old fashioned glass or heavy tumbler. Add sugar to taste and crush the lime and sugar together with a pestle or the end of a wooden spoon (be sure to crush the pieces pulp side up or too much bitter lime oil will be released from the zest). Add the cachaca, stir to mix and add the ice cubes. Stir again and serve.
Note: Cachaca, a spirit distilled from sugarcane, is one of the most popular drinks in Brazil. Some of the better varieties are available in larger liquor stores in the USA.
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