We present the analysis of the first set of low-redshift Type Ia supernovae (SNe Ia) by the Carnegie Supernova Project. Well-sampled, high-precision optical (ugriBV) and near-infrared (NIR; YJHKs) light curves obtained in a well-understood photometric system are used to provide light-curve parameters, and ugriBVYJH template light curves. The intrinsic colors at maximum light are calibrated to compute optical--NIR color excesses for the full sample, thus allowing the properties of the reddening law in the host galaxies to be studied. A low value of Rv~1.7, is derived when using the entire sample of SNe. However, when the two highly reddened SNe in the sample are excluded, a value Galactic standard of Rv~3.2 is obtained. The colors of these two events are well matched by a reddening model due to circumstellar dust. The peak luminosities are calibrated using a two-parameter linear fit to the decline rates and the colors, or alternatively, the color excesses. In both cases, dispersions in absolute magnitude of 0.12--0.16 mag are obtained, depending on the filter-color combination. In contrast to the results obtained from color excesses, these fits give Rv~1--2, even when the two highly reddened SNe are excluded. This discrepancy suggests that, beyond the normal interstellar reddening produced in the host galaxies, there is an intrinsic dispersion in the colors of SNe Ia which is correlated with luminosity but independent of the decline rate. Finally, a Hubble diagram is produced by combining the results of the fits for each filter. The resulting scatter of 0.12 mag appears to be limited by peculiar velocities as evidenced by the strong correlation between the distance-modulus residuals among the different filters. The implication is that the actual precision of SN Ia distances is 3--4%.
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