  We present the measurement of Baryon Acoustic Oscillations (BAO) from the
Lyman-$\alpha$ (Ly$\alpha$) forest of high-redshift quasars with the first-year
dataset of the Dark Energy Spectroscopic Instrument (DESI). Our analysis uses
over $420\,000$ Ly$\alpha$ forest spectra and their correlation with the
spatial distribution of more than $700\,000$ quasars. An essential facet of
this work is the development of a new analysis methodology on a blinded
dataset. We conducted rigorous tests using synthetic data to ensure the
reliability of our methodology and findings before unblinding. Additionally, we
conducted multiple data splits to assess the consistency of the results and
scrutinized various analysis approaches to confirm their robustness. For a
given value of the sound horizon ($r_d$), we measure the expansion at $z_{\rm
eff}=2.33$ with 2\% precision, $H(z_{\rm eff}) = (239.2 \pm 4.8) (147.09~{\rm
Mpc} /r_d)$ km/s/Mpc. Similarly, we present a 2.4\% measurement of the
transverse comoving distance to the same redshift, $D_M(z_{\rm eff}) = (5.84
\pm 0.14) (r_d/147.09~{\rm Mpc})$ Gpc. Together with other DESI BAO
measurements at lower redshifts, these results are used in a companion paper to
constrain cosmological parameters.
