The Carnegie Supernova Project ( CSP ) is designed to measure the luminosity distance for Type Ia supernovae ( SNe Ia ) as a function of redshift , and to set observational constraints on the dark energy contribution to the total energy content of the Universe . The CSP differs from other projects to date in its goal of providing an I -band rest-frame Hubble diagram . Here we present the first results from near-infrared ( NIR ) observations obtained using the Magellan Baade telescope for SNe Ia with 0.1 < z < 0.7 . We combine these results with those from the low-redshift CSP at z < 0.1 ( 20 ) . In this paper , we describe the overall goals of this long-term program , the observing strategy , data reduction procedures , and treatment of systematic uncertainties . We present light curves and an I -band Hubble diagram for this first sample of 35 SNe Ia and we compare these data to 21 new SNe Ia at low redshift . These data support the conclusion that the expansion of the Universe is accelerating . When combined with independent results from baryon acoustic oscillations ( 15 ) , these data yield \Omega _ { m } = 0.27 \pm 0.02 ~ { } ( \mathrm { statistical } ) , and \Omega _ { DE } = 0.76 \pm 0.13 ~ { } ~ { } ( \mathrm { statistical } ) \pm 0.09 ~ { } ( \mathrm { % systematic } ) , for the matter and dark energy densities , respectively . If we parameterize the data in terms of an equation of state , w ( with no time dependence ) , assume a flat geometry , and combine with baryon acoustic oscillations , we find that w = -1.05 \pm 0.13 ~ { } ( \mathrm { statistical } ) \pm 0.09 ~ { } ( \mathrm { systematic } ) . The largest source of systematic uncertainty on w arises from uncertainties in the photometric calibration , signaling the importance of securing more accurate photometric calibrations for future supernova cosmology programs . Finally , we conclude that either the dust affecting the luminosities of SNe Ia has a different extinction law ( R _ { V } = 1.8 ) than that in the Milky Way ( where R _ { V } = 3.1 ) , or that there is an additional intrinsic color term with luminosity for SNe Ia , independent of the decline rate . Understanding and disentangling these effects is critical for minimizing the systematic uncertainties in future SN Ia cosmology studies .