We study density profiles of an ideal Fermi gas and observe Pauli suppression of density fluctuations (atom shot noise) for cold clouds deep in the quantum degenerate regime. Strong suppression is observed for probe volumes containing more than 10,000 atoms. Measuring the level of suppression provides sensitive thermometry at low temperatures. After this method of sensitive noise measurements has been validated with an ideal Fermi gas, it can now be applied to characterize phase transitions in strongly correlated many-body systems.PACS numbers: 03.75. Ss, 05.30.Fk, 67.85.Lm Systems of fermions obey the Pauli exclusion principle. Processes that would require two fermions to occupy the same quantum state are suppressed. In recent years, several classic experiments have directly observed manifestations of Pauli suppression in Fermi gases. Antibunching and the suppression of noise correlations are a direct consequence of the forbidden double occupancy of a quantum state. Such experiments were carried out for electrons [1][2][3], neutral atoms [4,5], and neutrons [6]. In principle, such experiments can be done with fermions at any temperature, but in practice low temperatures increase the signal. A second class of (two-body) Pauli suppression effects, the suppression of collisions, requires a temperature low enough such that the de Broglie wavelength of the fermions becomes larger than the range of the interatomic potential and p-wave collisions freeze out. Experiments observed the suppression of elastic collisions [7,8] and of clock shifts in radio frequency spectroscopy [9,10].Here we report on the observation of Pauli suppression of density fluctuations, a many-body phenomenon which occurs only at even lower temperatures in the quantum degenerate regime, where the Fermi gas is cooled below the Fermi temperature and the low lying quantum states are occupied with probabilities close to one. In contrast, an ideal Bose gas close to quantum degeneracy shows enhanced density fluctuations [11].The development of a technique to sensitively measure density fluctuations was motivated by the connection between density fluctuations and compressibility through the fluctuation dissipation theorem. In this paper, we validate our technique for determining the compressibility by applying it to the ideal Fermi gas. In future work, it could be extended to interesting many-body phases in optical lattices which are distinguished by their incompressibility [12]. These include the band insulator, Mott insulator, and also the antiferromagnet for which spin fluctuations, i.e. fluctuations of the difference in density between the two spin states, are suppressed.Until now, sub-Poissonian number fluctuations of ultracold atoms have been observed only for small clouds of bosons with typically a few hundred atoms [13][14][15][16] and directly [17,18] or indirectly [19] for the bosonic Mott in-