Given an unambiguous context-free grammar G and an integer n > 0, we give two algorithms for generating words derived from the grammar uniformly at random (u.a.r.). The first algorithm generates a word u.a.r. in O(n²) time using a data structure of size O(n). The second algorithm generates a word u.a.r. in O(n) time using a data structure of size O(n²). Both algorithms make use of a preprocessing stage, where a data structure is built to facilitate the later generation of words uniformly at random. The algorithms are analyzed in a unit-cost model where the grammar is considered as a constant, and a weighted probabilistic choice among m outcomes is computed in O(m) time. When the unit-cost memory assumption is replaced by a logarithmic-cost measure, we give a simple approximation scheme that reduces the bit complexity of the above space bounds by a factor of Θ(log n/n), while generating words with probability N^-1(1 + O(n^-k)) for any integer k >0, where N is the number of derivable words of length n.