  Hybrid (exotic) mesons, which are important predictions of quantum
chromodynamics (QCD), are states of quarks and anti-quarks bound by excited
gluons. First principle lattice study of such states would help us understand
the role of ``dynamical'' color in low energy QCD and provide valuable
information for experimental search for these new particles. In this paper, we
apply both improved gluon and quark actions to the hybrid mesons, which might
be much more efficient than the previous works in reducing lattice spacing
error and finite volume effect. Quenched simulations were done at $\beta=2.6$
and on a $\xi=3$ anisotropic $12^3\times36$ lattice using our PC cluster. We
obtain $2013 \pm 26 \pm 71$ MeV for the mass of the $1^{-+}$ hybrid meson
${\bar q}qg$ in the light quark sector, and $4369 \pm 37 \pm 99$Mev in the
charm quark sector; the mass splitting between the $1^{-+}$ hybrid meson ${\bar
c}c g$ in the charm quark sector and the spin averaged S-wave charmonium mass
is estimated to be $1302 \pm 37 \pm 99$ MeV. As a byproduct, we obtain $1438
\pm 32 \pm 57$ MeV for the mass of a P-wave $1^{++}$ ${\bar u}u$ or ${\bar d}d$
meson and $1499 \pm 28 \pm 65$ MeV for the mass of a P-wave $1^{++}$ ${\bar
s}s$ meson, which are comparable to their experimental value 1426 MeV for the
$f_1(1420)$ meson. The first error is statistical, and the second one is
systematical. The mixing of the hybrid meson with a four quark state is also
discussed.
