We present a measurement of the baryon acoustic oscillation (BAO) scale at redshift $z=2.35$ from the three-dimensional correlation of Lyman-$alpha$ (Ly$alpha$) forest absorption and quasars. The study uses 266,590 quasars in the redshift range $1.77<z<3.5$ from the Sloan Digital Sky Survey (SDSS) Data Release 14 (DR14). The sample includes the first two years of observations by the SDSS-IV extended Baryon Oscillation Spectroscopic Survey (eBOSS), providing new quasars and re-observations of BOSS quasars for improved statistical precision. Statistics are further improved by including Ly$alpha$ absorption occurring in the Ly$beta$ wavelength band of the spectra. From the measured BAO peak position along and across the line of sight, we determined the Hubble distance $D_{H}$ and the comoving angular diameter distance $D_{M}$ relative to the sound horizon at the drag epoch $r_{d}$: $D_{H}(z=2.35)/r_{d}=9.20pm 0.36$ and $D_{M}(z=2.35)/r_{d}=36.3pm 1.8$. These results are consistent at $1.5sigma$ with the prediction of the best-fit spatially-flat cosmological model with the cosmological constant reported for the Planck (2016) analysis of cosmic microwave background anisotropies. Combined with the Ly$alpha$ auto-correlation measurement presented in a companion paper, the BAO measurements at $z=2.34$ are within $1.7sigma$ of the predictions of this model.
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