We present constraints on the parameters of the ACDM cosmological model in the presence of massive neutrinos, using the one-dimensional Ly alpha forest power spectrum obtained with the Baryon Oscillation Spectroscopic Survey (BOSS) of the Sloan Digital Sky Survey (SDSS) by Palanque-Delabrouille et al. [1], complemented by additional cosmological probes. The interpretation of the measured Ly alpha spectrum is done using a second-order Taylor expansion of the simulated power spectrum. BOSS Ly alpha data alone provide better bounds than previous Ly alpha results, but are still poorly constraining, especially for the sum of neutrino masses Sigma m(v), for which we obtain an upper bound of 1.1 eV (95% CL), including systematics for both data and simulations. Ly alpha constraints on ACDM parameters and neutrino masses are compatible with CMB bounds from the Planck collaboration [2]. Interestingly, the combination of Ly alpha with CMB data reduces the uncertainties significantly, due to very different directions of degeneracy in parameter space, leading to the strongest cosmological bound to date on the total neutrino mass, Sigma m(v) < 0.15 eV at 95% CL (with a best-fit in zero). Adding recent BAO results further tightens this constraint to Sigma m(v) < 0.14 eV at 95% CL. This bound is nearly independent of the statistical approach used, and of the different combinations of CMB and BAO data sets considered in this paper in addition to Ly alpha. Given the measured values of the two squared mass differences Delta m(2), this result tends to favor the normal hierarchy scenario against the inverted hierarchy scenario for the masses of the active neutrino species.