We create a sample of spectroscopically identified galaxies with $z < 0.2$ from the Sloan Digital Sky Survey (SDSS) Data Release 7, covering 6813 deg$^2$. Galaxies are chosen to sample the highest mass haloes, with an effective bias of 1.5, allowing us to construct 1000 mock galaxy catalogs (described in Paper II), which we use to estimate statistical errors and test our methods. We use an estimate of the gravitational potential to reconstruct the linear density fluctuations, enhancing the Baryon Acoustic Oscillation (BAO) signal in the measured correlation function and power spectrum. Fitting to these measurements, we determine $D_{V}(z_{rm eff}=0.15) = (664pm25)(r_d/r_{d,{rm fid}})$ Mpc; this is a better than 4 per cent distance measurement. This fills the gap in BAO distance ladder between previously measured local and higher redshift measurements, and affords significant improvement in constraining the properties of dark energy. Combining our measurement with other BAO measurements from BOSS and 6dFGS galaxy samples provides a 15 per cent improvement in the determination of the equation of state of dark energy and the value of the Hubble parameter at $z=0$ ($H_0$). Our measurement is fully consistent with the Planck results and the $Lambda$CDM concordance cosmology, but increases the tension between Planck$+$BAO $H_0$ determinations and direct $H_0$ measurements.
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