We report on our recent determination of the Boltzmann constant, k(B), by means of Doppler broadening thermometry. This relatively new method of primary gas thermometry was implemented by using a pair of offset-frequency locked extended-cavity diode lasers at 1.39 mu m, to probe a particular vibration-rotation transition of the (H2O)-O-18 molecule. Adopting a rather sophisticated and extremely refined line shape model in the spectral analysis procedure, we were able to determine the Doppler width from high-quality absorption spectra with unprecedented accuracy. Our spectroscopic determination of kB exhibits a combined (type A plus type B) uncertainty of 24 parts over 10(6). The complete uncertainty budget is presented and discussed.