We present the baryon acoustic oscillation (BAO) measurements with the Lyman-α (Lyα) forest from the second data release (DR2) of the Dark Energy Spectroscopic Instrument (DESI) survey. Our BAO measurements include both the autocorrelation of the Lyα forest absorption observed in the spectra of high-redshift quasars and the cross-correlation of the absorption with the quasar positions. The total sample size is approximately a factor of 2 larger than the DR1 dataset, with forest measurements in over 820,000 quasar spectra and the positions of over 1.2 million quasars. We describe several significant improvements to our analysis in this paper, and two supporting papers describe improvements to the synthetic datasets that we use for validation and how we identify damped Lyα absorbers. Our main result is that we have measured the BAO scale with a statistical precision of 1.1% along and 1.3% transverse to the line of sight, for a combined precision of 0.65% on the isotropic BAO scale at zeff=2.33. This excellent precision, combined with recent theoretical studies of the BAO shift due to nonlinear growth, motivated us to include a systematic error term in Lyα BAO analysis for the first time. We measure the ratios DH(zeff)/rd=8.632±0.098±0.026 and DM(zeff)/rd=38.99±0.52±0.12, where DH=c/H(z) is the Hubble distance, DM is the transverse comoving distance, rd is the sound horizon at the drag epoch, and we quote both the statistical and the theoretical systematic uncertainty. The companion paper presents the BAO measurements at lower redshifts from the same dataset and the cosmological interpretation.