  Ultra-low-mass axions are a viable dark matter candidate and may form a
coherently oscillating classical field. Nuclear spins in experiments on Earth
might couple to this oscillating axion dark-matter field, when propagating on
Earth's trajectory through our Galaxy. This spin coupling resembles an
oscillating pseudo-magnetic field which modulates the spin precession of
nuclear spins. Here we report on the null result of a demonstration experiment
searching for a frequency modulation of the free spin-precession signal of
\magHg in a \SI{1}{\micro\tesla} magnetic field. Our search covers the axion
mass range $10^{-16}~\textrm{eV} \lesssim m_a \lesssim 10^{-13}~\textrm{eV}$
and achieves a peak sensitivity to the axion-nucleon coupling of $g_{aNN}
\approx 3.5 \times 10^{-6}~\textrm{GeV}^{-1}$.
