  We study the neutral Higgs sector of the minimal supersymmetric standard
model (MSSM) with explicit CP violation at the one-loop level. We take into
account the one-loop contributions by the top quark, the stop quarks, the
bottom quark, the sbottom quarks, the tau lepton, the stau leptons, the $W$
boson, the charged Higgs boson, the charginos, the $Z$ boson, the neutral Higgs
bosons, and the neutralinos. The production cross sections of the neutral Higgs
boson are calculated to the leading order. The processes in our consideration
are divided in two groups: the Higgs-strahlung and gluon fusion processes
accessible at the CERN Large Hadron Collider (LHC), and the vector boson fusion
and Higgs-strahlung processes accessible at the $e^+e^-$ International Linear
Collider (ILC). In particular, we investigate the dependence of these processes
on the CP phase arising from the U(1) factor of the gaugino mass in the
neutralino mass matrix. We show that the cross sections of these processes vary
by the range of 3% $-$ 19 % as the CP phase changes from zero to $\pi$.
