  Her X-1, one of the brightest and best studied X-ray binaries, shows a
cyclotron resonant scattering feature (CRSF) near 37 keV. This makes it an
ideal target for detailed study with the Nuclear Spectroscopic Telescope Array
(NuSTAR), taking advantage of its excellent hard X-ray spectral resolution. We
observed Her X-1 three times, coordinated with Suzaku, during one of the high
flux intervals of its 35d super-orbital period. This paper focuses on the shape
and evolution of the hard X-ray spectrum. The broad-band spectra can be fitted
with a powerlaw with a high-energy cutoff, an iron line, and a CRSF. We find
that the CRSF has a very smooth and symmetric shape, in all observations and at
all pulse-phases. We compare the residuals of a line with a Gaussian optical
depth profile to a Lorentzian optical depth profile and find no significant
differences, strongly constraining the very smooth shape of the line. Even
though the line energy changes dramatically with pulse phase, we find that its
smooth shape does not. Additionally, our data show that the continuum is only
changing marginally between the three observations. These changes can be
explained with varying amounts of Thomson scattering in the hot corona of the
accretion disk. The average, luminosity-corrected CRSF energy is lower than in
past observations and follows a secular decline. The excellent data quality of
NuSTAR provides the best constraint on the CRSF energy to date.
