We have developed two independent methods to measure 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. The determination of the noise level in the data spectra was subject to a novel 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, large signal-to-noise ratio, and good spectral resolution. The power spectra measured using either approach are in good agreement over all twelve redshift bins from \( = 2.2\) to \( = 4.4\), and scales from 0.001 \(\rm(km/s)^{-1}\) to \(0.02 \rm(km/s)^{-1}\). We determine 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 \(\Lambda\)CDM model and using a constraint on \(H_0\) that encompasses measurements based on the local distance ladder and on CMB anisotropies, we infer \(\sigma_8 =0.83\pm0.03\) and \(n_s= 0.97\pm0.02\) based on \ion{H}{i} absorption in the range \(2.1<z<3.7\).