This paper presents the Planck 2013 likelihood, a complete statistical description of the two-point correlation function of the CMB temperature fluctuations that accounts for all known relevant uncertainties, both instrumental and astrophysical in nature. We use this likelihood to derive our best estimate of the CMB angular power spectrum from Planck over three decades in multipole moment, ℓ, covering 2 ≤ ℓ ≤ 2500. The main source of uncertainty at ℓ ≲ 1500 is cosmic variance. Uncertainties in small-scale foreground modelling and instrumental noise dominate the error budget at higher ℓs. For ℓ< 50, our likelihood exploits all Planck frequency channels from 30 to 353 GHz, separating the cosmological CMB signal from diffuse Galactic foregrounds through a physically motivated Bayesian component separation technique. At ℓ ≥ 50, we employ a correlated Gaussian likelihood approximation based on a fine-grained set of angular cross-spectra derived from multiple detector combinations between the 100, 143, and 217 GHz frequency channels, marginalising over power spectrum foreground templates. We validate our likelihood through an extensive suite of consistency tests, and assess the impact of residual foreground and instrumental uncertainties on the final cosmological parameters. We find good internal agreement among the high-ℓ cross-spectra with residuals below a few μK2 at ℓ ≲ 1000, in agreement with estimated calibration uncertainties. We compare our results with foreground-cleaned CMB maps derived from all Planck frequencies, as well as with cross-spectra derived from the 70 GHz Planck map, and find broad agreement in terms of spectrum residuals and cosmological parameters. We further show that the best-fit ΛCDM cosmology is in excellent agreement with preliminary PlanckEE and TE polarisation spectra. We find that the standard ΛCDM cosmology is well constrained by Planck from the measurements at ℓ ≲ 1500. One specific example is the spectral index of scalar perturbations, for which we report a 5.4σ deviation from scale invariance, ns = 1. Increasingthe multipole range beyond ℓ ≃ 1500 does not increase our accuracy for the ΛCDM parameters, but instead allows us to study extensions beyond the standard model. We find no indication of significant departures from the ΛCDM framework. Finally, we report a tension between the Planck best-fit ΛCDM model and the low-ℓ spectrum in the form of a power deficit of 5-10% at ℓ ≲ 40, with a statistical significance of 2.5-3σ. Without a theoretically motivated model for this power deficit, we do not elaborate further on its cosmological implications, but note that this is our most puzzling finding in an otherwise remarkably consistent data set.