We present the measurement of Baryon Acoustic Oscillations (BAO) from the Lyman- a (Lya) forest of high-redshift quasars with the first-year dataset of the Dark Energy Spectroscopic Instrument (DESI). Our analysis uses over 420 000 Lya 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 ( rd), we measure the expansion at zeff = 2.33 with 2% precision, H(zeff) = (239.2 +/- 4.8) (147.09Mpc/ rd) km/s/Mpc. Similarly, we present a 2.4% measurement of the transverse comoving distance to the same redshift, DM (zeff) = (5.84 +/- 0.14) ( rd/147.09Mpc) Gpc. Together with other DESI BAO measurements at lower redshifts, these results are used in a companion paper to constrain cosmological parameters.