  The anisotropy splitting in the ground state of the single-molecule magnet
[Mn4O3Br(OAc)3(dbm)3] is studied by inelastic neutron scattering as a function
of hydrostatic pressure. This allows a tuning of the anisotropy and thus the
energy barrier for slow magnetisation relaxation at low temperatures. The value
of the negative axial anisotropy parameter $D_{\rm cluster}$ changes from
-0.0627(1) meV at ambient to -0.0603(3) meV at 12 kbar pressure, and in the
same pressure range the height of the energy barrier between up and down spins
is reduced from 1.260(5) meV to 1.213(9) meV. Since the $\rm Mn-Br$ bond is
significantly softer and thus more compressible than the $\rm Mn-O$ bonds,
pressure induces a tilt of the single ion Mn$^{3+}$ anisotropy axes, resulting
in the net reduction of the axial cluster anisotropy.
