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 199Hg in a magnetic field. Our search covers the axion mass range 10^{-16} \textrm{eV} \lesssim m_a \lesssim 10^{-13} \textrm{eV} 10 − 16 eV ≲ m a ≲ 10 − 13 eV and achieves a peak sensitivity to the axion-nucleon coupling of g_{aNN} \approx 3.5 \times 10^{-6} \textrm{GeV}^{-1} g a N N ≈ 3.5 × 10 − 6 GeV − 1 .