  We present cosmological results from the measurement of baryon acoustic
oscillations (BAO) in galaxy, quasar and Lyman-$\alpha$ forest tracers from the
first year of observations from the Dark Energy Spectroscopic Instrument
(DESI), to be released in the DESI Data Release 1. DESI BAO provide robust
measurements of the transverse comoving distance and Hubble rate, or their
combination, relative to the sound horizon, in seven redshift bins from over 6
million extragalactic objects in the redshift range $0.1<z<4.2$. DESI BAO data
alone are consistent with the standard flat $\Lambda$CDM cosmological model
with a matter density $\Omega_\mathrm{m}=0.295\pm 0.015$. Paired with a BBN
prior and the robustly measured acoustic angular scale from the CMB, DESI
requires $H_0=(68.52\pm0.62)$ km/s/Mpc. In conjunction with CMB anisotropies
from Planck and CMB lensing data from Planck and ACT, we find
$\Omega_\mathrm{m}=0.307\pm 0.005$ and $H_0=(67.97\pm0.38)$ km/s/Mpc. Extending
the baseline model with a constant dark energy equation of state parameter $w$,
DESI BAO alone require $w=-0.99^{+0.15}_{-0.13}$. In models with a time-varying
dark energy equation of state parametrized by $w_0$ and $w_a$, combinations of
DESI with CMB or with SN~Ia individually prefer $w_0>-1$ and $w_a<0$. This
preference is 2.6$\sigma$ for the DESI+CMB combination, and persists or grows
when SN~Ia are added in, giving results discrepant with the $\Lambda$CDM model
at the $2.5\sigma$, $3.5\sigma$ or $3.9\sigma$ levels for the addition of
Pantheon+, Union3, or DES-SN5YR datasets respectively. For the flat
$\Lambda$CDM model with the sum of neutrino mass $\sum m_\nu$ free, combining
the DESI and CMB data yields an upper limit $\sum m_\nu < 0.072$ $(0.113)$ eV
at 95% confidence for a $\sum m_\nu>0$ $(\sum m_\nu>0.059)$ eV prior. These
neutrino-mass constraints are substantially relaxed in models beyond
$\Lambda$CDM. [Abridged.]
