  We use 1169 Pan-STARRS supernovae (SNe) and 195 low-$z$ ($z < 0.1$) SNe Ia to
measure cosmological parameters. Though most Pan-STARRS SNe lack spectroscopic
classifications, in a previous paper (I) we demonstrated that photometrically
classified SNe can be used to infer unbiased cosmological parameters by using a
Bayesian methodology that marginalizes over core-collapse (CC) SN
contamination. Our sample contains nearly twice as many SNe as the largest
previous SN Ia compilation. Combining SNe with Cosmic Microwave Background
(CMB) constraints from Planck, we measure the dark energy equation of state
parameter $w$ to be -0.989$\pm$0.057 (stat$+$sys). If $w$ evolves with redshift
as $w(a) = w_0 + w_a(1-a)$, we find $w_0 = -0.912 \pm 0.149$ and $w_a =$
-0.513$\pm$0.826. These results are consistent with cosmological parameters
from the Joint Lightcurve Analysis and the Pantheon sample. We try four
different photometric classification priors for Pan-STARRS SNe and two
alternate ways of modeling CC SN contamination, finding that no variant gives a
$w$ differing by more than 2% from the baseline measurement. The systematic
uncertainty on $w$ due to marginalizing over CC SN contamination,
$\sigma_w^{\textrm{CC}} = 0.012$, is the third-smallest source of systematic
uncertainty in this work. We find limited (1.6$\sigma$) evidence for evolution
of the SN color-luminosity relation with redshift, a possible systematic that
could constitute a significant uncertainty in future high-$z$ analyses. Our
data provide one of the best current constraints on $w$, demonstrating that
samples with $\sim$5% CC SN contamination can give competitive cosmological
constraints when the contaminating distribution is marginalized over in a
Bayesian framework.
