We report on a new implementation of Doppler broadening thermometry based on precision absorption spectroscopy by means of a pair of offset-frequency locked extended-cavity diode lasers at 1.39 mu m. The method consists in the highly accurate observation of the shape of the 4(4,1) -> 4(4,0) line of the (H2O)-O-18 nu(1) + nu(3) band, in a water vapor sample at thermodynamic equilibrium. A sophisticated and extremely refined spectral analysis procedure is adopted for the retrieval of the Doppler width as a function of the gas pressure, taking into account the Dicke narrowing effect, the speed dependence of relaxation rates, and the physical correlation between velocity-changing and dephasing collisions. A spectroscopic determination of the Boltzmann constant with a combined (type A and type B) uncertainty of 24 parts over 10(6) is reported. This is the best result obtained so far by means of an optical method. Our determination is in agreement with the recommended CODATA value.