  We present constraints on cosmological parameters using number counts as a
function of redshift for a sub-sample of 189 galaxy clusters from the Planck SZ
(PSZ) catalogue. The PSZ is selected through the signature of the
Sunyaev--Zeldovich (SZ) effect, and the sub-sample used here has a
signal-to-noise threshold of seven, with each object confirmed as a cluster and
all but one with a redshift estimate. We discuss the completeness of the sample
and our construction of a likelihood analysis. Using a relation between mass
$M$ and SZ signal $Y$ calibrated to X-ray measurements, we derive constraints
on the power spectrum amplitude $\sigma_8$ and matter density parameter
$\Omega_{\mathrm{m}}$ in a flat $\Lambda$CDM model. We test the robustness of
our estimates and find that possible biases in the $Y$--$M$ relation and the
halo mass function are larger than the statistical uncertainties from the
cluster sample. Assuming the X-ray determined mass to be biased low relative to
the true mass by between zero and 30%, motivated by comparison of the observed
mass scaling relations to those from a set of numerical simulations, we find
that $\sigma_8=0.75\pm 0.03$, $\Omega_{\mathrm{m}}=0.29\pm 0.02$, and
$\sigma_8(\Omega_{\mathrm{m}}/0.27)^{0.3} = 0.764 \pm 0.025$. The value of
$\sigma_8$ is degenerate with the mass bias; if the latter is fixed to a value
of 20% we find $\sigma_8(\Omega_{\mathrm{m}}/0.27)^{0.3}=0.78\pm 0.01$ and a
tighter one-dimensional range $\sigma_8=0.77\pm 0.02$. We find that the larger
values of $\sigma_8$ and $\Omega_{\mathrm{m}}$ preferred by Planck's
measurements of the primary CMB anisotropies can be accommodated by a mass bias
of about 40%. Alternatively, consistency with the primary CMB constraints can
be achieved by inclusion of processes that suppress power on small scales
relative to the $\Lambda$CDM model, such as a component of massive neutrinos
(abridged).
