  We analyse the Baryon Acoustic Oscillation (BAO) signal of the final Baryon
Oscillation Spectroscopic Survey (BOSS) data release (DR12). Our analysis is
performed in Fourier-space, using the power spectrum monopole and quadrupole.
The dataset includes $1\,198\,006$ galaxies over the redshift range $0.2 < z <
0.75$. We divide this dataset into three (overlapping) redshift bins with the
effective redshifts $\zeff = 0.38$, $0.51$ and $0.61$. We demonstrate the
reliability of our analysis pipeline using N-body simulations as well as $\sim
1000$ MultiDark-Patchy mock catalogues, which mimic the BOSS-DR12 target
selection. We apply density field reconstruction to enhance the BAO
signal-to-noise ratio. By including the power spectrum quadrupole we can
separate the line-of-sight and angular modes, which allows us to constrain the
angular diameter distance $D_A(z)$ and the Hubble parameter $H(z)$ separately.
We obtain two independent $1.6\%$ and $1.5\%$ constraints on $D_A(z)$ and
$2.9\%$ and $2.3\%$ constraints on $H(z)$ for the low ($\zeff=0.38$) and high
($\zeff=0.61$) redshift bin, respectively. We obtain two independent $1\%$ and
$0.9\%$ constraints on the angular averaged distance $D_V(z)$, when ignoring
the Alcock-Paczynski effect. The detection significance of the BAO signal is of
the order of $8\sigma$ (post-reconstruction) for each of the three redshift
bins. Our results are in good agreement with the Planck prediction within
$\Lambda$CDM. This paper is part of a set that analyses the final galaxy
clustering dataset from BOSS. The measurements and likelihoods presented here
are combined with others in~\citet{Alam2016} to produce the final cosmological
constraints from BOSS.
