LetF(x)be an absolutely continuous distribution having a density functionf(x)with respect to the Lebesgue measure. The Shannon entropy is defined asH(f) = -\int f(x) \ln f(x) dx. In this correspondence we propose, based on a random sampleX_{1}, \cdots , X_{n}generated fromF, a nonparametric estimate ofH(f)given by\hat{H}(f) = -(l/n) \sum_{i = 1}^{n} \In \hat{f}(x), where\hat{f}(x)is the kernel estimate offdue to Rosenblatt and Parzen. Regularity conditions are obtained under which the first and second mean consistencies of\hat{H}(f)are established. These conditions are mild and easily satisfied. Examples, such as Gamma, Weibull, and normal distributions, are considered.