We have developed two independent methods for measuring the one-dimensional power spectrum of the transmitted flux in the Lyman-α forest. The first method is based on a Fourier transform and the second on a maximum-likelihood estimator. The two methods are independent and have different systematic uncertainties. Determination of the noise level in the data spectra was subject to a new treatment, because of its significant impact on the derived power spectrum. We applied the two methods to 13 821 quasar spectra from SDSS-III/BOSS DR9 selected from a larger sample of over 60 000 spectra on the basis of their high quality, high signal-to-noise ratio (S/N), and good spectral resolution. The power spectra measured using either approach are in good agreement over all twelve redshift bins from 〈z〉 = 2.2 to 〈z〉 = 4.4, and scales from 0.001 km s -1 to 0.02 km s -1 . We determined the methodological andinstrumental systematic uncertainties of our measurements. We provide a preliminary cosmological interpretation of our measurements using available hydrodynamical simulations. The improvement in precision over previously published results from SDSS is a factor 2-3 for constraints on relevant cosmological parameters. For a ΛCDM model and using a constraint on H 0 that encompasses measurements based on the local distance ladder and on CMB anisotropies, we infer σ 8 = 0.83 ± 0.03 and n s = 0.97 ± 0.02 based on Hi absorption in the range 2.1 < z < 3.7. © ESO, 2013.