We report on gamma-ray observations of the supernova remnant (SNR) RX J0852.0--4622 with the Large Area Telescope (LAT) on board the Fermi Gamma-ray Space Telescope. In the Fermi-LAT data, we find a spatially extended source at the location of the SNR. The extension is consistent with the SNR size seen in other wavelengths such as X-rays and TeV gamma rays, leading to the identification of the gamma-ray source with the SNR. The spectrum is well described as a power law with a photon index of Delta *G = 1.85 ? 0.06 (stat)+0.18 -- 0.19 (sys), which smoothly connects to the H.E.S.S. spectrum in the TeV energy band. We discuss the gamma-ray emission mechanism based on multiwavelength data. The broadband data can be fit well by a model in which the gamma rays are of hadronic origin. We also consider a scenario with inverse Compton scattering of electrons as the emission mechanism of the gamma rays. Although the leptonic model predicts a harder spectrum in the Fermi-LAT energy range, the model can fit the data considering the statistical and systematic errors.