We performed the DC-magnetization and neutron scattering experiments under pressure {it P} for a pressure-induced superconductor UGe$_2$. We found that the magnetic moment is enhanced at a characteristic temperature {it T}$^{*}$ in the ferromagnetic state, where {it T}$^{*}$ is smaller than a Curie temperature {it T}$_{rm C}$. This enhancement becomes remarkable in the vicinity of {it P}$_{rm C}^{*}$ = 1.20 GPa, where {it T}$^{*}$ becomes 0 K and the superconducting transition temperature {it T}$_{rm SC}$ shows a maximum. The characteristic temperature {it T}$^{*}$, which decreases with increasing pressure, also depends on the magnetic field.
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