  We present a new cosmological analysis of the galaxy clusters in the Planck
MMF3 cosmology sample with a cosmic microwave background (CMB) lensing
calibration of the cluster masses. As demonstrated by Planck, galaxy clusters
detected via the Sunyaev-Zel'dovich (SZ) effect offer a powerful way to
constrain cosmological parameters such as $\Omega_{\mathrm{m}}$ and $\sigma_8$.
Determining the absolute cluster mass scale is, however, difficult, and some
recent calibrations have yielded cosmological constraints in apparent tension
with constraints in the $\Lambda$CDM model derived from the power spectra of
the primary CMB anisotropies. In order to calibrate the absolute mass scale of
the full Planck cluster sample, we remeasure the masses of all 433 clusters
through their weak lensing signature in the CMB temperature anisotropies as
measured by Planck. We perform a joint Bayesian analysis of the cluster counts
and masses taking as input the estimated cluster masses, SZ signal-to-noise
ratios, and redshifts. Our analysis properly accounts for selection effects in
the construction of the cluster sample. We find $\sigma_8
\left(\Omega_{\mathrm{m}}/0.33\right)^{0.25} = 0.765 \pm 0.035$ and
$1-b_{\mathrm{SZ}} = 0.71 \pm 0.10$, where the mass bias factor
$1-b_{\mathrm{SZ}}$ relates cluster mass to the SZ mass that appears in the
X-ray-calibrated cluster scaling relations. We find no evidence for tension
with the Planck primary CMB constraints on $\Lambda$CDM model parameters.

Cosmology and Nongalactic Astrophysics