We present a measurement of baryon acoustic oscillations (BAO) in the cross-correlation of quasars with the Ly$alpha$-forest flux-transmission at a mean redshift $z=2.40$. The measurement uses the complete SDSS-III data sample: 168,889 forests and 234,367 quasars from the SDSS Data Release DR12. In addition to the statistical improvement on our previous study using DR11, we have implemented numerous improvements at the analysis level allowing a more accurate measurement of this cross-correlation. We also developed the first simulations of the cross-correlation allowing us to test different aspects of our data analysis and to search for potential systematic errors in the determination of the BAO peak position. We measure the two ratios $D_{H}(z=2.40)/r_{d} = 9.01 pm 0.36$ and $D_{M}(z=2.40)/r_{d} = 35.7 pm 1.7$, where the errors include marginalization over the non-linear velocity of quasars and the metal - quasar cross-correlation contribution, among other effects. These results are within $1.8sigma$ of the prediction of the flat-$Lambda$CDM model describing the observed CMB anisotropies. We combine this study with the Ly$alpha$-forest auto-correlation function [2017A&A...603A..12B], yielding $D_{H}(z=2.40)/r_{d} = 8.94 pm 0.22$ and $D_{M}(z=2.40)/r_{d} = 36.6 pm 1.2$, within $2.3sigma$ of the same flat-$Lambda$CDM model.
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