  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.
