  We calculate the next-to-leading order contribution to the masses of the
heavy baryons in the bound state approach for baryons containing a heavy quark.
These $1/N_C$ corrections arise when states of good spin and isospin are
generated from the background soliton of the light meson fields. Our study is
motivated by the previously established result that light vector meson fields
are required for this soliton in order to reasonably describe the spectrum of
both the light and the heavy baryons. We note that the inclusion of light
vector mesons significantly improves the agreement of the predicted hyperfine
splitting with experiment. A number of aspects of this somewhat complicated
calculation are discussed in detail.
