  We present cosmological constraints from the sample of Type Ia supernovae (SN
Ia) discovered during the full five years of the Dark Energy Survey (DES)
Supernova Program. In contrast to most previous cosmological samples, in which
SN are classified based on their spectra, we classify the DES SNe using a
machine learning algorithm applied to their light curves in four photometric
bands. Spectroscopic redshifts are acquired from a dedicated follow-up survey
of the host galaxies. After accounting for the likelihood of each SN being a SN
Ia, we find 1635 DES SNe in the redshift range $0.10<z<1.13$ that pass quality
selection criteria sufficient to constrain cosmological parameters. This
quintuples the number of high-quality $z>0.5$ SNe compared to the previous
leading compilation of Pantheon+, and results in the tightest cosmological
constraints achieved by any SN data set to date. To derive cosmological
constraints we combine the DES supernova data with a high-quality external
low-redshift sample consisting of 194 SNe Ia spanning $0.025<z<0.10$. Using SN
data alone and including systematic uncertainties we find $\Omega_{\rm
M}=0.352\pm 0.017$ in flat $\Lambda$CDM. Supernova data alone now require
acceleration ($q_0<0$ in $\Lambda$CDM) with over $5\sigma$ confidence. We find
$(\Omega_{\rm M},w)=(0.264^{+0.074}_{-0.096},-0.80^{+0.14}_{-0.16})$ in flat
$w$CDM. For flat $w_0w_a$CDM, we find $(\Omega_{\rm
M},w_0,w_a)=(0.495^{+0.033}_{-0.043},-0.36^{+0.36}_{-0.30},-8.8^{+3.7}_{-4.5})$.
Including Planck CMB data, SDSS BAO data, and DES $3\times2$-point data gives
$(\Omega_{\rm M},w)=(0.321\pm0.007,-0.941\pm0.026)$. In all cases dark energy
is consistent with a cosmological constant to within $\sim2\sigma$. In our
analysis, systematic errors on cosmological parameters are subdominant compared
to statistical errors; paving the way for future photometrically classified
supernova analyses.
