  In this paper, we present significant progress performed on an experiment
dedicated to the determination of the Boltzmann constant, k, by accurately
measuring the Doppler absorption profile of a line in a gas of ammonia at
thermal equilibrium. This optical method based on the first principles of
statistical mechanics is an alternative to the acoustical method which has led
to the unique determination of k published by the CODATA with a relative
accuracy of 1.7 ppm. We report on the first measurement of the Boltzmann
constant by laser spectroscopy with a statistical uncertainty below 10 ppm,
more specifically 6.4 ppm. This progress results from improvements in the
detection method and in the statistical treatment of the data. In addition, we
have recorded the hyperfine structure of the probed saQ(6,3) rovibrational line
of ammonia by saturation spectroscopy and thus determine very precisely the
induced 4.36 (2) ppm broadening of the absorption linewidth. We also show that,
in our well chosen experimental conditions, saturation effects have a
negligible impact on the linewidth. Finally, we draw the route to future
developments for an absolute determination of with an accuracy of a few ppm.
