The Carnegie Supernova Project (CSP) is a five-year survey being carried out at the Las Campanas Observatory to obtain high-quality light curves of ~100 low-redshift Type Ia supernovae (SNe Ia) in a well-defined photometric system. Here we present the first release of photometric data that contains the optical light curves of 35 SNe Ia, and near-infrared light curves for a subset of 25 events. The data comprise 5559 optical (ugriBV) and 1043 near-infrared (Y JHKs ) data points in the natural system of the Swope telescope. Twenty-eight SNe have pre-maximum data, and for 15 of these, the observations begin at least 5 days before B maximum. This is one of the most accurate data sets of low-redshift SNe Ia published to date. When completed, the CSP data set will constitute a fundamental reference for precise determinations of cosmological parameters, and serve as a rich resource for comparison with models of SNe Ia.