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 1 $\mu$T magnetic field. Our search covers the axion mass range $10^{-16}$ eV $\lesssim m_a \lesssim 10^{-13}$ eV and achieves a peak sensitivity to the axion-nucleon coupling of $g_{aNN} \approx 3.5 \times 10^{-6}$ GeV$^{-1}$.
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