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 $\Omega m$ 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 $\sigma 8(\Omega m/0.33)0.25=0.765\pm 0.035$ 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.