Local density fluctuations and density profiles of a Fermi gas are measured in situ and analyzed. In the quantum degenerate regime, the weakly interacting 6Li gas shows a suppression of the density fluctuations compared to the nondegenerate case, where atomic shot noise is observed. This manifestation of antibunching is a direct result of the Pauli principle and constitutes a local probe of quantum degeneracy. We analyze our data using the predictions of the fluctuation-dissipation theorem and the local density approximation, demonstrating a fluctuation-based temperature measurement.