We have developed two independent methods for measuring the one-dimensional power spectrum of the transmitted flux in the Lyman-alpha 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 and instrumental 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 ACDM model and using a constraint on H-0 that encompasses measurements based on the local distance ladder and on CMB anisotropies, we infer sigma(8) = 0.83 +/- 0.03 and n(s) = 0.97 +/- 0.02 based on HI absorption in the range 2.1 < z < 3.7.
