We present gamma-ray observations with the LAT on board the Fermi Gamma-Ray Telescope of the nearby radio galaxy Centaurus~A. The previous EGRET detection is confirmed, and the localization is improved using data from the first 10 months of Fermi science operation. In previous work, we presented the detection of the lobes by the LAT; in this work, we concentrate on the gamma-ray core of Cen~A. Flux levels as seen by the LAT are not significantly different from that found by EGRET, nor is the extremely soft LAT spectrum ($G=2.67pm0.10_{stat}pm0.08_{sys}$ where the photon flux is $Phipropto E^{-G}$). The LAT core spectrum, extrapolated to higher energies, is marginally consistent with the non-simultaneous HESS spectrum of the source. The LAT observations are complemented by simultaneous observations from Suzaku, the Swift Burst Alert Telescope and X-ray Telescope, and radio observations with the Tracking Active Galactic Nuclei with Austral Milliarcsecond Interferometry (TANAMI) program, along with a variety of non-simultaneous archival data from a variety of instruments and wavelengths to produce a spectral energy distribution (SED). We fit this broadband data set with a single-zone synchrotron/synchrotron self-Compton model, which describes the radio through GeV emission well, but fails to account for the non-simultaneous higher energy TeV emission observed by HESS from 2004-2008. The fit requires a low Doppler factor, in contrast to BL Lacs which generally require larger values to fit their broadband SEDs. This indicates the $g$-ray emission originates from a slower region than that from BL Lacs, consistent with previous modeling results from Cen~A. This slower region could be a slower moving layer around a fast spine, or a slower region farther out from the black hole in a decelerating flow.
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