CUORE-0 is a cryogenic detector that uses an array of tellurium dioxide bolometers to search for neutrinoless double-beta decay of 130Te . We present the first data analysis with 7.1kg·y of total TeO2 exposure focusing on background measurements and energy resolution. The background rates in the neutrinoless double-beta decay region of interest (2.47 to 2.57MeV ) and in the α background-dominated region (2.70 to 3.90MeV ) have been measured to be 0.071±0.011 and 0.019±0.002counts/(keV·kg·y) , respectively. The latter result represents a factor of 6 improvement from a predecessor experiment, Cuoricino. The results verify our understanding of the background sources in CUORE-0, which is the basis of extrapolations to the full CUORE detector. T...