  Spectroscopy has an illustrious history delivering serendipitous discoveries
and providing a stringent testbed for new physical predictions, including
applications from trace materials detection, to understanding the atmospheres
of stars and planets, and even constraining cosmological models. Reaching
fundamental-noise limits permits optimal extraction of spectroscopic
information from an absorption measurement. Here we demonstrate a
quantum-limited spectrometer that delivers high-precision measurements of the
absorption lineshape. These measurements yield a ten-fold improvement in the
accuracy of the excited-state (6P$_{1/2}$) hyperfine splitting in Cs, and
reveals a breakdown in the well-known Voigt spectral profile. We develop a
theoretical model that accounts for this breakdown, explaining the observations
to within the shot-noise limit. Our model enables us to infer the thermal
velocity-dispersion of the Cs vapour with an uncertainty of 35ppm within an
hour. This allows us to determine a value for Boltzmann's constant with a
precision of 6ppm, and an uncertainty of 71ppm.
