  We study the spectra of the hybrid mesons containing one heavy quark
($q\bar{Q}g$) within the framework of QCD sum rules in the heavy quark limit.
The derived sum rules are stable with the variation of the Borel parameter
within their corresponding working ranges. The extracted binding energy for the
heavy hybrid doublets $H^h(S^h)$ and $M^h(T^h)$ is almost degenerate. We also
calculate the pionic couplings between these heavy hybrid and the conventional
heavy meson doublets using the light-cone QCD sum rule method. The extracted
coupling constants are rather small as a whole. With these couplings we make a
rough estimate of the partial widths of these pionic decay channels.
