  We measure the imprint of primordial baryon acoustic oscillations (BAO) in
the correlation function of Ly$\alpha$ absorption in quasar spectra from the
Baryon Oscillation Spectroscopic Survey (BOSS) and the extended BOSS (eBOSS) in
Data Release 14 (DR14) of the Sloan Digital Sky Survey (SDSS)-IV. In addition
to 179,965 spectra with absorption in the Lyman-$\alpha$ (Ly$\alpha$) region,
we use, for the first time, Ly$\alpha$ absorption in the Lyman-$\beta$ region
of 56,154 spectra. We measure the Hubble distance, $D_H$, and the comoving
angular diameter distance, $D_M$, relative to the sound horizon at the drag
epoch $r_d$ at an effective redshift $z=2.34$. Using a physical model of the
correlation function outside the BAO peak, we find $D_H(2.34)/r_d=8.86\pm 0.29$
and $D_M(2.34)/r_d=37.41\pm 1.86$, within 1$\sigma$ from the flat-$\Lambda$CDM
model consistent with CMB anisotropy measurements. With the addition of
polynomial "broadband" terms, the results remain within one standard deviation
of the CMB-inspired model. Combined with the quasar-Ly$\alpha$
cross-correlation measurement presented in a companion paper Blomqvist19, the
BAO measurements at $z=2.35$ are within 1.7$\sigma$ of the predictions of this
model.
