  On the arcminute angular scales probed by Planck, the CMB anisotropies are
gently perturbed by gravitational lensing. Here we present a detailed study of
this effect, detecting lensing independently in the 100, 143, and 217GHz
frequency bands with an overall significance of greater than 25sigma. We use
the temperature-gradient correlations induced by lensing to reconstruct a
(noisy) map of the CMB lensing potential, which provides an integrated measure
of the mass distribution back to the CMB last-scattering surface. Our lensing
potential map is significantly correlated with other tracers of mass, a fact
which we demonstrate using several representative tracers of large-scale
structure. We estimate the power spectrum of the lensing potential, finding
generally good agreement with expectations from the best-fitting LCDM model for
the Planck temperature power spectrum, showing that this measurement at z=1100
correctly predicts the properties of the lower-redshift, later-time structures
which source the lensing potential. When combined with the temperature power
spectrum, our measurement provides degeneracy-breaking power for parameter
constraints; it improves CMB-alone constraints on curvature by a factor of two
and also partly breaks the degeneracy between the amplitude of the primordial
perturbation power spectrum and the optical depth to reionization, allowing a
measurement of the optical depth to reionization which is independent of
large-scale polarization data. Discarding scale information, our measurement
corresponds to a 4% constraint on the amplitude of the lensing potential power
spectrum, or a 2% constraint on the RMS amplitude of matter fluctuations at
z~2.
